Chapter 4
NC Commands

Mathematica 3.0 has a lovely graphical user interface which uses Palettes. Mathematica Palettes display the most important commands and prompt the user. We have such a Palette for NCAlgebra and NCGB which contain most of the commands in this chapter. See the TEAR OFF Section in the back for a picture of the Mma Palettes for NCAlgebra and NCGB. To pop up this Palette, open a notebook, load NCAlgebra or NCGB, then open the file NCPalette.nb. If you are in a directory containing the file NCPalette.nb you can open it directly from a notebook.

4.1 Manipulating an expression

4.1.1 ExpandNonCommutativeMultiply[expr]

Aliases: NCE,NCExpand

Description: ExpandNonCommutativeMultiply[expr] expands out NonCommutativeMultiply’s in expr. It is the noncommutative generalization of the Mma Expand command.

Arguments: expr is an algebraic expression.

Comments / Limitations: None

4.1.2 NCCollect[expr, aListOfVariables]

Aliases: NCC

Description: NCCollect[expr,aListOfV ariables] collects terms of expression expr according to the elements of aListOfV ariables and attempts to combine them using a particular list of rules called rulesCollect. NCCollect is weaker than NCStrongCollect in that first-order and second-order terms are not collected together. NCCollect uses NCDecompose, and then NCStrongCollect, and then NCCompose.

Arguments: expr is an algebraic expression. aListOfV ariables is a list of variables.

Comments / Limitations: While NCCollect[expr,x] always returns mathematically correct expressions, it may not collect x from as many terms as it should. If expr has been expanded in the previous step, the problem does not arise. If not, the pattern match behind NCCollect may not get entirely inside of every factor where x appears.

4.1.3 NCStrongCollect[expr, aListOfVariables]

Aliases: NCSC

Description: It collects terms of expression expr according to the elements of aListOfV ariables and attempts to combine them using the particular list of rules called rulesCollect. In the noncommutative case, the Taylor expansion, and hence the collect function, is not uniquely specified. This collect function often collects too much and while mathematically correct is often stronger than you want. For example, x will factor out of terms where it appears both linearly a quadratically thus mixing orders.

Arguments: expr is an algebraic expression. aListOfV ariables is a list of variables.

Comments / Limitations: Not well documented.

4.1.4 NCCollectSymmetric[expr]

Aliases: NCCSym

Description: None

Arguments: expr is an algebraic expression.

Comments / Limitations: None

4.1.5 NCTermsOfDegree[expr,aListOfVariables,indices]

Aliases: None

Description: NCTermsOfDegree[expr,aListOfV ariables,indices] returns an expression such that each term is homogeneous of degree given by the indices in the variables of aListOfV ariables. For example, NCTermsOfDegree[x**y**x+x**x**y+x**x+ x**w,{x,y},indices] returns x**x**y +x**y **x if indices = {2, 1}, return x**w if indices = {1, 0}, return x **x if indices = {2, 0} and returns 0 otherwise. This is like Mathematica’s Coefficient command, but for the noncommuting case. However, it actually gives the terms and not the coefficients of the terms.

Arguments: expr is an algebraic expression, aListOfV ariables is a list of variables and indices is a list of positive integers which is the same length as aList.

Comments / Limitations: Not available before NCAlgebra 1.0

4.1.6 NCSolve[expr1==expr2,var]

Aliases: None

Description: NCSolve[expr1 == expr2,var] solves some simple equations which are linear in the unknown var. Note that in the noncommutative case, many equations such as Lyapunov equations cannot be solved for an unknown. This obviously is a limitation on the NCSolve command.

Arguments: expr1 and expr2 are Mathematica expressions. var is a single variable.

Comments / Limitations: See description.

4.1.7 Substitute[expr,aListOfRules,(Optional On)]

Aliases: Sub

Description: It repeatedly replaces one symbol or sub-expression in the expression by another expression as specified by the rule. (See Wolfram’s Mathematica 2.* book page 54.) More recently, we wrote the Transform command (§4.1.11) which apprears to be better.

Arguments: expr is an algebraic expression. aListOfRules is a single rule or list of rules specifying the substitution to be made. On = save rules to Rules.temp, temporarily over-riding SaveRules[Off]. ‘Off’ cannot over-ride SaveRules[On].

Comments / Limitations: The symbols /. and //. are often used in Mathematica as methods for substituting one expression for another. This method of substitution often does not work when the expression to be substituted is a subexpression within a (noncommutative) product. This Substitute command is the noncommutative analogue to //.

4.1.8 SubstituteSymmetric[expr, aListOfRules, (optional On)]

Aliases: SubSym

Description: When a rule specifies that a → b, then SubSym also makes the replacement tp[a] → tp[b].

Arguments: expr is an algebraic expression. aListOfRules is a single rule or list of rules specifying the substitution to be made. On = save rules to Rules.temp, temporarily over-rides SaveRules[Off]. ’Off’ can not over-ride SaveRules[On].

Comments / Limitations: None

4.1.9 SubstituteSingleReplace[expr, aListOfRules, (optional On)]

Aliases: SubSingleRep

Description: Replaces one symbol or sub-expression in the expression by another expression as specified by the rule. (See Wolfram’s Mathematica 2.* page 54.)

Arguments: expr is an algebraic expression. aListOfRules is a single rule or list of rules specifying the substitution to be made. On = save rules to Rules.temp, temporarily over-rides SaveRules[Off]. ‘Off’ can not over-ride SaveRules[On].

Comments / Limitations: The symbols /. and //. are often used in Mathematica as methods for substituting one expression for another. This method of substitution often does not work when the expression to be substituted is a subexpression within a (noncommutative) product. This Substitute command is the noncommutative analogue to /.

4.1.10 SubstituteAll[expr, aListOfRules, (optional On)]

Aliases: SubAll

Description: For every rule a → b, SubAll also replaces,

Arguments: expr is an algebraic expression. aListOfRules is a single rule or list of rules specifying the substitution to be made. On = save rules to Rules.temp, temporarily over-riding SaveRules[Off]. ’Off’ can not over-ride SaveRules[On].

Comments / Limitations: None

4.1.11 Transform[expr,aListOfRules]

Aliases: Transform

Description: None

Arguments: Transform is essentially a more efficient version of Substitute. It has the same functionality as Substitute.

Comments / Limitations: expr is an algebraic expression. aListOfRules is a single rule or list of rules specifying the substitution to be made.

Beware: Transform only applies rules once rather than repeatedly.

4.1.12 GrabIndeterminants[ aListOfPolynomialsOrRules]

Aliases: none

Description: GrabIndeterminants[L] returns the indeterminates found in the list of (noncommutative) expressions or rules L. For example, GrabIndeterminants[ { x**Inv[x]**x + Tp[Inv[x+a]], 3 + 4 Inv[a]**b**Inv[a] + x }] returns

Arguments: aListOfPolynomialsOrRules is a list of (noncommutative) expressions or rules.

Comments / Limitations:

4.1.13 GrabVariables[ aListOfPolynomialsOrRules ]

Aliases: none

Description: GrabVariables[ aListOfPolynomialsOrRules ] returns the variables found in the list of (noncommutative) expressions or rules aListOfPolynomialsOrRules. It is similar to the Mathematica command Variables[] which takes as an argument a list of polynomials in commutative variables or functions of variables. For example, returns

Arguments: aListOfPolynomialsOrRules is a list of (noncommutative) expressions or rules.

Comments / Limitations:

4.1.14 NCBackward[expr]

Aliases: NCB

Description: It applies the rules

Arguments: expr is an algebraic expression.

Comments / Limitations: None

4.1.15 NCForward[expr]

Aliases: NCF

Description: It applies the rules

Arguments: expr is an algebraic expression.

Comments / Limitations: None

4.1.16 NCMonomial[expr]

Aliases: None

Description: NCMonomail changes the look of an expression by replacing nth integer powers of the NonCommutative variable x, with the product of n copies of x. For example, NCMonomial[2x2 + 5x4] evaluates to 2x **x + 5x **x **x **x and NCMonomial[(x2) **z **x] evaluates to x **x **z **x.

Arguments: Any noncommutative expression.

Comments / Limitations: The program greatly eases the task of typing in polynomials. For example, instead of typing x = x**x**x**x**x**x**x**x**x**x**x**x**y**x**x, one can type x = NCMono[(x12)**y**(x2)]. NCMono expands only integer exponents. This program will be (or has been, depending on the version of code which you have) superseded by NCMonomial and NCUnMonomial. NCMonomial implements the same functionality as NCMonomial and NCUnMonomial reverses the process. Caution: Mathematica treats x**y2 as (x**y)2 and so to have Mathematica acknowledge x**y2 then input x**(y2) exactly. This has nothing to do with NCAlgebra or NCMonomial.

4.1.17 NCUnMonomial[expr]

Aliases: None

Description: NCUnMonomial reverses what NCMonomial does. NCUnMonomial changes the look of an expression by replacing a product of n copies of x with xⁿ. For example, NCUnMonomial[2x **x + 5x **x **x **x] evaluates to 2x2 + 5x4 and NCUnMonomial[x **x **z **x] evaluates to (x2) **z **x.

Arguments: Any noncommutative expression.

Comments / Limitations: See NCMonomial. NCAlgebra does not effectively manipulate expressions involving powers (such as (x2))

4.2 Simplification

This area is under developement so stronger commands will appear in later versions. What we mean by simplify is not in the spirit of Mathematica’s Simplify. They tend to factor expressions so that the expressions become very short. We expand expressions apply rules to the expressions which incorporate special relations the entries satisfy. Then we rely on cancelation of terms. The theoretical background lies in noncommutative Gröbner basis theory, and the rules we are implementing come from papers of Helton, Stankus and Wavrik [IEEE TAC March 1998].

The commands in this section are designed to simplify polynomials in a, b, inv[S - a **b], inv[S - b **a], inv[S - a], inv[S - b] and a few slightly more complicated inverses.

The commands in order of strength are NCSR, NCS1R, NCS2R. Of course, for a stronger the command, more rules get applied and so the command takes longer to run.

First, NCS1R normalizes inv[S - a **b] to S^-1 * inv[1 - (a **b)∕S] provided S is a commutative expression (only works for numbers S in version 0.2 of NCAlgebra). The following list of rules are applied.

(0) inv[-1 + a] →-inv[1 - a]
(1) inv[1 - a] (a - b) inv[1 - b] → inv[1 - a] - inv[1 - b]
(2) inv[1 - ab] inv[b] → inv[1 - ba] a + inv[b]
(3) inv[1 - ab] ab → inv[1 - ab] - 1
(4) abinv[1 - ab] → inv[1 - ab] - 1
(5) inv[c] inv[1 - cb] → inv[1 - bc] inv[c]
(6) b inv[1 - ab] → inv[1 - ba]b

The command NCS2R increases the range of expressions to include inv[poly], but the reductions for each of these inverses is considerably less powerful than for the case of inv[1 - ab].

An example: if expr = a**inv[a + b] + inv[c-a] **(a-c) + inv[c + d] **(c + d + e), then the first reduction using the list of rules in NCSR gives a**inv[a + b] + inv[c + d] **einv[a] **(a-b) **inv[b] and the second reduction gives inv[b] - inv[a] which is the output from NCSR[expr].

NCSimplify0Rational is an old attempt at simplification. We do not use it much.

4.2.1 NCSimplifyRational[ expr ], NCSimplify1Rational[ expr ], and NCSimplify2Rational[ expr ]

Aliases: NCSR

Description: The objective is to simplify expressions which include polynomials and inverses of very simple polynomials. These work by appling a collection of relations implemented as rules to expr. The core of NCSimplifyRational is NCSimplify1Rational and NCSimplify2Rational; indeed roughly NCSimplifyRational[expr] = NCSimplify1Rational[NCSimplify2Rational[expr]] together with some NCExpand’s. NCSimplify1Rational[expr] contains one set of rules while NCSimplify2Rational[expr] contains another.

Arguments: expr is an algebraic expression.

Comments / Limitations: Works only for a specialized class of functions.

4.2.2 NCSimplify1Rational[expr]

Aliases: NCS1R

Description: It applies a collection of relations implemented as rules to expr. The goal is to simplify expr.

Arguments: expr is an algebraic expression.

Comments / Limitations: WARNING: NCS1R does not first do an ExpandNonCommutativeMultiply. Therefore, it may be the case that one can miss some simplification if expr is not expanded out. The solution, of course, is to call ExpandNonCommutativeMultiply before calling NCS1R. ExpandNonCommutativeMultiply is called from NCSR.

 First, NCS1R normalizes inv[S -a**b] to S^-1 *inv[1 - (a**b)∕S] provided S is s a commutative expression (only works for numbers S in version 0.2 of NCAlgebra). The the following list of rules are applied.
(0) inv[-1 + a] →-inv[1 - a]
(1) inv[1 - a] (a - b) inv[1 - b] → inv[1 - a] - inv[1 - b]
(2) inv[1 - ab] inv[b] → inv[1 - ba] a + inv[b]
(3) inv[1 - ab] ab → inv[1 - ab] - 1
(4) abinv[1 - ab] → inv[1 - ab] - 1
(5) inv[c] inv[1 - cb] → inv[1 - bc] inv[c]
(6) b inv[1 - ab] → inv[1 - ba]b

In the notation of papers [HW], [HSW], these rules implement a superset of the union of the Gröbner basis for EB and the Gröbner basis for RESOL.

4.2.3 NCSimplify2Rational[expr]

Aliases: NCS2R

Description: You need this for expressions involving inv[ polynomial ] where the polynomial is not of the form SId - X **Y

Arguments: expr is an algebraic expression.

Comments / Limitations: If the polynomial is too complicated, this may not help very much.

4.3 Vector Differentiation

4.3.1 DirectionalD[expr, aVariable, h]

Aliases: DirD

Description: Takes the Directional Derivative of expression expr with respect to the variable aV ariable in direction h.

Arguments: expr is an expression containing var. aV ariable is a variable. h is the direction which the derivative is taken in.

Comments / Limitations: None.

4.3.2 Grad[expr, aVariable]

Aliases: Grad, NEVER USE Gradient

Description: Grad[expr,aV ariable] takes the gradient of expression expr with respect to the variable aV ariable. Quite useful for computations with quadratic Hamiltonians in H^∞ control. BEWARE Gradient calls the Mma gradient and makes a mess.

Arguments: expr is an expression containing var. aV ariable is a variable.

Comments / Limitations: This only works reliably for quadratic expressions. It is not even correct on all of these. For example, Grad[a **x + a **tp[x],x] returns 2tp[a]. The reason is fundamental mathematics, not programming. If a is a row vector and x is a column vector, then a **x is a number, but a **tp[x] is not.

4.3.3 CriticalPoint[expr, aVariable]

Aliases: Crit, Cri

Description: It finds the value of aV ariable which makes the gradient of the expression expr with respect to the variable aV ariable equal to 0.

Arguments: expr is an expression containing aV ariable. aV ariable is a variable.

Comments / Limitations: Uses the Grad and NCSolve functions. Both Grad and NCSolve are severely limited. Therefore, the CriticalPoint command has a very limited range of applications.

4.3.4 NCHessian[afunction, {X₁,H₁},…,{X_k,H_k} ]

Aliases: None.

Description: NCHessian[afunction,{X₁,H₁},{X₂,H₂},…,{X_k,H_k} ]
computes the Hessian of a afunction of noncommutting variables and coefficients. The Hessian recall is the second derivative. Here we are computing the noncommutative directional derivative of a noncommutative function. Using repeated calls to DirectionalD, the Hessian of afunction is computed with respect to the variables X₁,X₂,…,X_k and the search directions H₁ , H₂ , … , H_k. The Hessian Γ of a function Γ is defined by One can easily show that the second derivative of a hereditary symmetric noncommutative rational function Γ with respect to one variable X has the form where A_ℓ, B_ℓ, and C_ℓ are functions of X determined by Γ. (An analogous expression holds for more variables.) The Hessian will always be quadratic with respect to . (A noncommutative polynomial in variables H₁, H₂, … , H_k, is said to be quadratic if each monomial in the polynomial expression is of order two in the variables H₁, H₂, …, H_k.)

Arguments: afunction is a function of the variables X₁,X₂,…,X_k. The Hessian will be computed with respect to the search directions H₁ , H₂ , … , H_k.
For example, suppose F(x,y) = x + x **y + y **x. Then,
NCHessian[F,{x,h},{y,k}] gives 2h**k + 2k **h As another example, if G(x,y,z) = inv[y] + z **x, then NCHessian[G,{x,h},{y,k},{z,i}] gives 2i **h + 2inv[y] **k * *inv[y] **k **inv[y].
The results of NCHessian can be factored into the form v^tMv by calling NCMatrixofQuadratic. (see NCMatrixofQuadratic).

Comments / Limitations: None.

4.4 Block Matrix Manipulation

By block matrices we mean matrices with noncommuting entries.

The Mathematica convention for handling vectors is tricky.

is a 1×3 matrix or a row vector

is a 3×1 matrix or a column vector

is a vector but NOT A MATRIX. Indeed whether it is a row or column vector depends on the context. DON’T USE IT. Always remember to use TWO curly brackets on your vectors or there will probably be trouble.

As of NCAlgebra version 3.2 one can handle block matrix manipulation two different ways. One is the old way as described below where you use the command MatMult[A, B] to multiply block matrices A and B and tpMat[A] to take transposes. The other way is much more pleasing though still a little risky. First you use the NCGuts[] with the Options NCStrongProduct1 → True to change ** to make block matrices multiply corectly. Further invoke the Option NCStrongProduct2 → True to strengthen the power of **. Now one does not have to use MatMult and tpMat; just use ** and tp instead it recognizes matrix sizes and multiplies correctly.

4.4.1 MatMult[x, y, …]

Aliases: MM

Description: MatMult multiplies matrices. The Mathematica code executed for MatMult[x,y] is Inner[ NonCommutativeMultiply, x, y, Plus];

Arguments: x is a block matrix, and y is a block matrix.

Comments / Limitations: MatMult can take any number of input parameters. For example, MatMult[a, b, c, d] will give the same result as MatMult[a, MatMult[b, MatMult[c, d]] ].

4.4.2 ajMat[u]

Aliases: None

Description: ajMat[u] returns the transpose of the block matrix u. The Mathematica code is Transpose[Map[aj[#]&,u, 2]];

Arguments: u is a block m × n matrix.

Comments / Limitations: None

4.4.3 coMat[u]

Aliases: None

Description: coMat[u] returns the transpose of the block matrix u. The Mathematica code is [Map[co[#]&,u, 2]];

Arguments: u is a block m × n matrix

Comments / Limitations: None

4.4.4 tpMat[u]

Aliases: None

Description: tpMat[u] returns the transpose of the block matrix u. The Mathematica is Transpose[Map[tp[#]&,u, 2]];

Arguments: u is a block m × n matrix

Comments / Limitations: None

4.4.5 NCMToMatMult[expr]

Aliases: None

Description: Sometimes one develops an expression in which ** occurs between matrices. This command takes all ** and converts them to MatMult. The Mathematica code executed is expr//.NonCommutativeMultiply → MatMult;

Arguments: expr is an algebraic expression. This and its inverse (TimesToNCM) are important in manipulating block matrices. One can use instead of this command, since that is all that this command amounts to.

Comments / Limitations: None

4.4.6 TimesToNCM[expr]

Aliases: TTNCM

Description: The Mathematica code executed is expr∕.Times → NonCommutativeMultiply

Arguments: expr is an algebraic expression.

Comments / Limitations: It changes commutative multiplication (Times) to NonCommutative multiplication.

4.4.7 Special Operations with Block Matrices

In 1999, we produced commands for LU decomposition and Cholesky decomposition of an inversion of matrices with noncommutative entries. These replace older commands GaussElimination[X] and invMat2[mat] for 2 × 2 block matrices which are no longer documented. The next 6 commands do that.

4.4.8 NCLDUDecomposition[aMatrix, Options]

Aliases: None.

Description: NCLDUDecomposition[X] yields the LDU decomposition for a square matrix X. It returns a list of four elements, namely L,D,U, and P such that PXP^T = LDU. The first element is the lower triangular matrix L, the second element is the diagonal matrix D, the third element is the upper triangular matrix U, and the fourth is the permutation matrix P (the identity is returned if no permutation is needed). As an option, it may also return a list of the permutations used at each step of the LDU factorization as a fifth element.

Suppose X is given by X = {{a,b, 0},{0,c,d},{a, 0,d}}. The command returns matrices, which in MatrixForm are:

As matrix X is 3×3, one can provide 2 permutation matrices. Let those permutations be given by l₁ = {3, 2, 1} and l₂ = {1, 3, 2}, that means:

just as in NCPermutationMatrix. The command

returns matrices, which in MatrixForm are:

It can be checked that P^T lo di up P = X:

Arguments: X is a square matrix n by n. The default Options are:
{Permutation → False, CheckDecomposition → False,
NCSimplifyPivots → False, StopAutoPermutation → False,
ReturnPermutation → False, Stop2by2Pivoting → False }. If permutation matrices are to be given, they should be provided as Permutation → {l₁, l₂, , l_n}, where each l_i is a list of integers (see the command NCPermutationMatrix[]). If CheckDecomposition is set to True, the function checks if PXP^T is identical to LDU. Where P = P₁P₂P_n, and each P_i is the permutation matrix associated with each l_i.

Often a prospective pivot will appear to be nonzero in Mathematica even though it reduces to zero. To ensure we are not pivoting with a convoluted form of zero, we simplify the pivot at each step. By default, NCLDUDecomposition converts the pivot from non-commutative to commutative and then simplifies the expression. If the commutative form of the pivot simplifies to zero, Mathematica scrolls down the diagonal looking for a pivot which does not simplify to zero. If all the diagonal entries simplify to zero utilizing the CommuteEverything[] command, the process is repeated using NCSimplifyRational.
This strategy is incorporated for two main reasons. One is that for large matrices it is much faster. Secondly, NCSimplifyRational does not always completely simplify complicated expressions. Setting NCSimplifyPivots → True bypasses CommuteEverything and immediately applies
NCSimplifyRational to each pivot. NCLDUDecomposition will automatically pivot if the current pivot at a particular iteration is zero. If the user utilized the Permutation option, then the permutation designated will be temporarily disregarded. However, NCLDUDecomposition will try and use the given permutation list for the next step. In this way,
NCLDUDecomposition follows the user permutation as closely as possible. If StopAutoPermutation → True, then NCLDUDecomposition will not automatically pivot and will strictly adhere to the user’s permutation, attempting to divide by zero if need be. This will allow the user to determine which permutations are not possible. Because NCLDUDecomposition will automatically pivot when necessary by default, the ReturnPermutation was created so that the permutation used in the decomposition can be returned to the user for further analysis if set to True.
To explain the last option it is somewhat necessary for the user to have an idea of how the pivoting strategy works. The permutations used are always symmetrically applied. Because of this, we can only place other diagonal elements in the (1,1) position. However, it is possible to place any off diagonal element in the (2,1) position. Thus our strategy is to pivot only with diagonal elements if possible. If all the diagonal elements are zero, then a permutation matrix is used to place a nonzero entry in the (2,1) position which will automaticaly place a nonzero entry in the (1,2) position if the matrix is symmetric. Then, instead of using the (1,1) entry as a pivot, the 2×2 submatrix starting in the (1,1) position is used as a block pivot. This has the effect of creating an LDU decomposition where D is a block diagonal matrix with 1×1 and 2×2 blocks along the diagonal. (Note: The pivots are precisely the diagonal entries of D.) Setting Stop2by2Pivoting → True will halt 2 × 2 block pivoting, returning instead, the remaining undecomposed block with zeros along the diagonal as a final block diagonal entry.

Comments / Limitations: NCLDUDecomposition automatically assumes invertible any expressions (pivot) it needs to be invertible. Also, the 2 × 2 pivoting strategy assumes that the matrix is symmetric in that it only ensures that the (2,1) entry is nonzero (assuming by symmetry that the (1,2) is also zero). The pivoting strategy chooses its pivots based upon the smallest leaf count invoking the Mathematica command LeafCount[]. It will choose the smallest nonzero diagonal element basing size upon the leaf count. This strategy is incorporated in an attempt to find the simplest LDU factorization possible. If a 2 × 2 pivot is used and ReturnPermutation is set to True then at the end of the permutation list returned will be the string 2by2 permutation.

4.4.9 NCAllPermutationLDU[aMatrix]

Aliases: None.

Description: NCAllPermutationLDU[aMatrix] returns the LDU decomposition of a matrix for all possible permutations. The code cycles through all possible permutations and calls NCLDUDecomposition for each one.

Arguments: aMatrix is a square matrix. The default options for NCAllPermutationLDU are: PermutationSelection → False, CheckDecomposition → False, NCSimplifyPivots → False, StopAutoPermutation → False, ReturnPermutation → False, Stop2by2Pivoting → False. All of these options have the same effect as in NCLDUDecomposition, except for PermutationSelection. PermutationSelection should be a list of numbers between 1 and the number of possible permutations. NCAllPermutationLDU will use this list to choose the permutations from its canonical list to decompose the matrix using NCLDUDecomposition. For example, PermutationSelection can be {1,…,n}.

Comments / Limitations: The output is a list of all successful outputs from NCLDUDecomposition. Note that some permutations may lead to a zero pivot in the process of doing the LDU decomposition. In that case, the LDU decomposition is not well defined, actually in Mathematica one gets a lot of ∞ signs, but this output will not be included in the list of successful outputs.

4.4.10 NCInverse[aSquareMatrix]

Aliases: None.

Description: NCInverse[m] gives a symbolic inverse of a matrix with noncommutative entries.

Arguments: m is an n × n matrix with noncommutative entries.

Comments / Limitations: This command is primarily used symbolically and is not guarenteed to work for any specific examples. Usually the elements of the inverse matrix (m^-1) are huge expressions. We recommend using NCSimplifyRational[NCInverse[m]] to improve the formula you get. In some cases, NCSimplifyRational[m^-1m] does not provide the identity matrix, even though it does equal the identity matrix. The formula we use for NCInverse[] comes from the LDU decomposition. Thus in principle it depends on the order chosen for pivoting even if the inverse of a matrix is unique.

4.4.11 NCPermutationMatrix[aListOfIntegers]

Aliases: None.

Description: NCPermutationMatrix[aListOfIntegers] returns the permutation matrix associated with the list of integers. It is just the identity matrix with its columns re-ordered.

Arguments: aListOfIntegers is an encoding which specifies where the 1’s occur in each column. e.g., aListOfintegers = {2, 4, 3, 1} represents the permutation matrix

Comments / Limitations: None.

4.4.12 NCMatrixToPermutation[aMatrix]

Aliases: None.

Description: NCMatrixToPermutation[aMatrix] returns the permutation associated with the permutation matrix, aMatrix. Basically, it is the inverse of NCPermutationMatrix.

Arguments: aMatrix must be matrix whose columns (or rows) can be permuted to yield the identity matrix. In other words, aMatrix must be a permutation matrix. For example, if m = {{0, 0, 0, 1},{1, 0, 0, 0},{0, 0, 1, 0},{0, 1, 0, 0}}, then NCPermutationMatrix[m] gives {2, 4, 3, 1}.

Comments / Limitations: None.

4.4.13 NCCheckPermutation[SizeOfMatrix, aListOfPermutations]

Aliases: None.

Description: If aListOfPermutations is consistent with the matrix size, SizeOfMatrix, then the output is valid permutation list. If not, the output is not valid permutation list.

Arguments: The size of a square matrix (an integer) and a list of permutations.

Comments / Limitations: If the SizeOfMatrix is n, then aListOfPermutations must be a list of n - 1 permutations of the integers 1 through n. Since this command is generally called within the context of NCLDUDecomposition the list of permutations must correspond to a list that can be used within the command.

4.4.14 Diag[aMatrix]

Aliases: None.

Description: Returns the elements of the diagonal of a matrix.

Arguments: None.

Comments / Limitations: The code is Flatten[MapIndexed[Part,m]].

4.4.15 Cascade[P, K]

Aliases: None

Description: Cascade[P,K] is the composition of P, K as is found is systems engineering.

Arguments: P is a 2×2 block matrix. K is a symbol.

Comments / Limitations: frequency response functions grow from this.

4.4.16 Chain[P]

Aliases: None

Description: Chain[P] returns the chain matrix arising from P as is found in systems engineering.

Arguments: P is a block 2×2 matrix.

Comments / Limitations: Chain[ ] assumes appropriate matrices are invertible.

4.4.17 Redheffer[P]

Aliases: None

Description: Redheffer[P] gives the inverse of chain.
Redheffer[Chain[P]] = P = Chain[Redheffer[P]].

Arguments: P is a block 2 × 2 matrix.

Comments / Limitations: Redheffer[P] assumes the invertiblity of the entries of P.

4.4.18 DilationHalmos[x]

Aliases: None

Description: DilationHalmos[x] gives block 2 × 2 matrix which is the Halmos dilation of x

Arguments: x is a symbol

Comments / Limitations: u = DilationHalmos[x] has the property u is unitary, that is, MatMult[u,tpMat[u]] == IdentityMatrix[2] and MatMult[tpMat[u],u] == IdentityMatrix[2].

4.4.19 SchurComplementTop[M]

Aliases: None

Description: SchurComplementTop[M] returns the Shur Complement of the top diagonal entry of a block 2 × 2 matrix M.

Arguments: M is a block 2 × 2 matrix.

Comments / Limitations: Assumes invertibility of a diagonal entry.

4.4.20 SchurComplementBtm[M]

Aliases: None

Description: SchurComplementBtm[M] returns the ShurComplement of the bottom diagonal entry of a block 2 × 2 matrix M.

Arguments: M is a block 2 × 2 matrix.

Comments / Limitations: Assumes invertibility of a diagonal entry.

4.5 Complex Analysis

4.5.1 A tutorial

The package in the file ComplexRules.m defines three objects:

∙ ComplexRules,    transformation rules

∙ ComplexCoordinates,     a function that applies rules to an expression.

∙ ComplexD[],     takes complex derivatives.

The ComplexRules package is for handling complex algebra and differentiation. The algebra part of ComplexRules has been pretty much superceeded by the standard Mathematica command ComplexExpand[] so we advise using that. Our complex differentiation is still quite useful. ComplexRules.m may not work well with ReIm.m, see the warning at the end of this note.

To rewrite this in terms of variables and their conjugates, apply the list of rules ComplexRules as follows

You can get the same result with the function ComplexCoordinates[]:

Suppose that you know that in the expression above, e ranges in the unit circle of the complex plane, and that w is real. To simplify you can do this:

Complex derivatives are easy to produce with ComplexD[]:

Here is a differentiation with respect to Conjugate[w]:

A mixed second order partial derivative is shown below:

Repeated differentiation is also possible:

Finally, we point out that it is possible that applying ComplexRules to an expression and applying ComplexCoordinates to it may yield different output (the same mathematically of course). Reason: ComplexCoordinates applies ComplexRules to the expression, in addition to a rule for transforming Abs[z] into Sqrt[ z Conjugate[z]]. Example:

ComplexD[] handles Abs[]² etc.:

ComplexD[] also handles Abs[]¹ but the answer does not look as pretty:

WARNING: The standard Mathematica package ReIm.m sets things so that expressions of complex variables “z” are rewritten in terms of Re[z], Im[z] (for example).

Compare this to the output of functions in the package ComplexRules.m, where the expressions of complex variables “z” are given in terms of z, Conjugate[z].

You may load both ReIm.m and ComplexRules.m, but keep in mind that the objectives of the packages conflict. Furthermore, programs that need ComplexRules to run will sometimes not work if ReIm.m has been loaded.

Mathematica can manipulate complex analysis via X + I Y where X and Y are commutative (e.g., numbers). However, it is often more convenient to calculate in terms of z and the conjugate of z. We implement a few commands in the file NCComplex.m. We discuss these commands below. One may also look at the file NCComplex.m for further documentation.

4.5.2 ComplexRules

Aliases: None

Description: ComplexRules is a set of replacement rules for writing expressions in terms of the variables and their complex conjugates. For example, use this with input containing numbers and variables, as well as operators/functions such as + - * ∕, Re[], Im[], Conjugate[], Exp[], Power[], Sin[], Cos[] and others. Apply the command expr//.ComplexRules. Try the following example:
Re[(1 + zw)²]² //.ComplexRules

Arguments: None

Comments / Limitations: This only works for expressions defined with the commutative multiplication.

4.5.3 ComplexCoordinates[expr]

Aliases: None

Description: ComplexCoordinates[expr] expands expr in terms of the variables and their complex conjugates. The difference between ComplexCoordinates[expr] and ComplexRules is in the case Abs[z]²//.ComplexRules. This case returns the same expression instead of z and Conjugate[z]. If you desire to use the latter expression, you can use ComplexCoordinates[expr]. This function replaces Abs[z] by Sqrt[z Conjugate[z]], after applying ComplexRules.

Arguments: expr is any expression with + - * ∕, Re[], Im[], Conjugate[], Exp[], Power[], Sin[], Cos[] and others

Comments / Limitations: This only works for expressions defined with the commutative multiplication.

4.5.4 ComplexD[expr, aVariable]

Aliases: None

Description: ComplexD[expr,aV ariable] calculates the derivative of the complex expression expr with respect to the “complex” variable aV ariable. You can also calculate the derivative with respect to Conjugate[aV ariable]. Try these examples:

ComplexD[Conjugate[Exp[z + 1∕Conjugate[z]]²],z];
ComplexD[Re[(1 + zw)²]²,w];
ComplexD[Abs[1∕(e² - 1) - z]²,z];
ComplexD[Conjugate[Exp[z + 1∕Conjugate[z]]²],Conjugate[z]];

Here is a second order derivative:

ComplexD[Conjugate[Exp[z + 1∕Conjugate[z]]²,z, 2];

Arguments: expr is a complex expression. aV ariable is the variable in which to take the derivative with respect to.

Comments / Limitations: This only works for expressions defined with the commutative multiplication.

4.6 Setting symbols to commute or not commute

4.6.1 SetNonCommutative[A, B, C, …]

Aliases: SNC, SetNC

Description: SetNonCommutative[A, B, C, …] sets all the symbols A, B, C, … to be noncommutative. The lower case letters a, b, c, … are assumed noncommutative by Mathematica default as are functions of noncommutative variables. The functions tp[] and aj[] are set noncommutative by NCAlgebra for any argument, commutative or noncommutative. We may change this.

Arguments: Symbols separated by commas

Comments / Limitations: None

4.6.2 CommuteEverything[expr]

Aliases: CE

Description: It changes NonCommutativeMultiply to Times in expr.

Arguments: expr is an algebraic expression.

Comments / Limitations: Very useful for getting ideas in the middle of a complicated calcuation. If expr has you baffled, type exprcom = CE[expr]. exprcom is commutative and therefore is easy to analyze. Now expr is uneffected, so you can get back to working on it armed with new ideas.

4.6.3 SetCommutative[a, b, c, …]

Aliases: None

Description: SetCommutative[a, b, c, …] sets all the symbols a, b, c, … to be commutative.

Arguments: Symbols separated by commas

Comments / Limitations: None

4.6.4 SetCommutingOperators[b,c]

Aliases: None

Description: SetCommutingOperators takes exactly two parameters. SetCommutingOperators[b, c] will implement the definitions which follow. They are in pseudo-code so that the meaning will not be obscured b ** c becomes c ** b if LeftQ[b, c]; and c ** b becomes b ** c if LeftQ[b, c]; ). See SetCommutingFunctions and LeftQ.

Arguments: b, c are symbols.

Comments / Limitations: NOTE: The above implementation will NOT lead to infinite loops.

 WARNING: If one says SetCommutingOperators[b, c] and then sets only LeftQ[c,b], then neither of the above rules will be executed. Therefore, one must remember the order of the two parameters in the statement. One obvious helpful habit would be to use alphabetical order (i.e., say SetCommutingOperators[a,b] and not the reverse).

4.6.5 LeftQ[expr]

Aliases: None

Description: See SetCommutingFunctions and SetCommutingOperators.

Arguments: expr is an algebraic expression.

Comments / Limitations: None

4.6.6 CommutativeQ[X]

Aliases: CQ

Description: CommutativeQ[X] is True if X is commutative, and False if X is noncommutative.

Arguments: X is a symbol.

Comments / Limitations: See the description of SetNonCommutative for the defaults.

4.6.7 CommutativeAllQ[expr]

Aliases: None

Description: CommutativeAllQ[expr] is True if expr does not have any non-commuting sub-expressions, and False otherwise.

Arguments: expr is an algebraic expression.

Comments / Limitations: None

4.7 Operations on elements in an algebra

4.7.1 inv[x]

Aliases: None

Description: Inverse – a ** inv[a]=inv[a] ** a=Id

Arguments: x is a symbol.

Comments / Limitations: Warning: NCAlgebra does not check that inv[x] exists or even that it makes sense (e.g. non-square matrices). This is the responsibility of the user.

4.7.2 invL[x]

Aliases: invL

Description: Left inverse – invL[a] ** a=Id

Arguments: x is a symbol

Comments / Limitations: Warning. NCAlgebra does not check that invL[x] exists. This is the responsibility of the user.

4.7.3 invR[x]

Aliases: invR

Description: invR[x] is the right inverse – a ** invR[a]=Id

Arguments: x is a symbol

Comments / Limitations: Warning. NCAlgebra does not check that invR[x] exists. This is the responsibility of the user.

4.7.4 invQ[x]

Aliases: None

Description: invQ[m] = True forces invR[m] and invL[m] to be rewritten as inv[m]

Arguments: x is an expression.

Comments / Limitations: We never use this command.

4.7.5 ExpandQ[inv]

Aliases: None

Description: If ExpandQ[inv] is set to True, an inverse of a product will be expanded to a product of inverses. If it is set to False, then a product of inverses will be rewritten to be a inverse of a product.

Arguments: inv

Comments / Limitations: None

4.7.6 ExpandQ[tp]

Aliases: None

Description: If ExpandQ[tp] is set to True, a transpose of a product will be expanded to a product of transposes. If it is set to False, then a product of transposes will be rewritten to be a transpose of a product.

Arguments: tp

Comments / Limitations: None

4.7.7 OverrideInverse

Aliases: None

Description: OverrideInverse is a variable which is either True or False.

Arguments: If OverrideInverse is set to True, then the replacement of invL and invR by inv (when x is invertible) is suppressed even if invQ is True. The default is False.

Comments / Limitations: None

4.7.8 aj[expr]

Aliases: None

Description: aj[expr] takes the adjoint of the expression expr. Note that basic laws like aj[a **b] = aj[b] **aj[a] are automatically executed.

Arguments: expr is an algebraic expression

Comments / Limitations: None

4.7.9 tp[expr]

Aliases: None

Description: tp[expr] takes the transpose of expression expr. Note that basic laws like tp[a **b] = tp[b] **tp[a] are automatically executed.

Arguments: expr is an algebraic expression

Comments / Limitations: None

4.7.10 co[expr]

Aliases: None

Description: co[expr] takes the complex conjugate of expr. Note basic laws like
co[a**b]=co[a]**co[b] and co[a]=aj[tp[a]]=tp[aj[a]]

Arguments: expr is an algebraic expression

Comments / Limitations: None

4.8 Convexity of a NC function

This chapter describes commands which do two things. One is compute the ”region” on which a noncommutative function is matrix convex. The other is take a noncommutative quadratic function variables H₁,H₂, etc and give a Gram representation for it, that is, represent it as

4.8.1 NCConvexityRegion[afunction,alistOfVars,opts]

Aliases: None.

Description: NCConvexityRegion[afunction,alistOfVars,opts] computes the ”region” on which afunction is matrix convex with respect to alistOfVars. It performs three main operations. First it computes the Hessian with respect to alistOfVars (see NCHessian). Then, using NCMatrixOfQuadratic, the Hessian is factored into the form v^tMv. Finally, depending on the option AllPermutation, either NCAllPermutationLDU or NCLDUDecomposition is called to compute the LDU factorization of M, the default being AllPermutation → NCLDUDecomposition. If D ends up being diagonal, then a list of the diagonal elements of D is returned. If D ends up being block diagonal with 2 × 2 blocks, then a message is printed out and the list: {{diagonalentries},{subdiagonalentries},{-subdiagonalentries}} is returned. The region of convexity of afunction with respect to alistOfVars equals the closure, in a certain sense, of the set of matrices which makes all diagonal entries positive. If there are non-zero subdiagonal entries, then afunction is typically not matrix convex on any open set. Options permit the user to select a range of different permutation matrices, thereby producing several possibly distinct diagonal matrices D.
EXAMPLE: NCConvexityRegion[x **y + y **x,{x,y}] gives:
  L**D**tp[L] gave non-trivial blocks, so the output list is:
  {{diagonal},{subdiagonal},{-subdiagonal}}
  {{{0, 0},{2},{-2}}}
While, NCConvexityRegion[x **y + y **x,{x,y},AllPermutation → True] gives:
  Middle matrix is size 2 X 2
  At most 2 permutations possible.
  {1}
  L**D**tp[L] gave non-trivial blocks, so the output list is:
  {{diagonal},{subdiagonal},{-subdiagonal}}
  {{{0, 0},{2},{-2}}}
  In both cases, NCHessian[x **y + y **x,{x,h},{y,k}] gives
  2h **k + 2k **h,
NCMatrixOfQuadratic[2h **k + 2k **h,{h,k}] gives
  {{{h,k}},{{0, 2},{2, 0}},{{h},{k}}},
and depending on if AllPermutation is set to True or False you have that either NCLDUDecomposition[{{0, 2},{2, 0}}] gives
  {{{1, 0},{0, 1}},{{0, 2},{2, 0}},{{1, 0},{0, 1}},{{1, 0},{0, 1}}}
or NCAllPermutationLDU[{{0, 2},{2, 0}}] gives
  {{{{1, 0},{0, 1}},{{0, 2},{2, 0}},{{1, 0},{0, 1}},{{1, 0},{0, 1}}},
  {{{1, 0},{0, 1}},{{0, 2},{2, 0}},{{1, 0},{0, 1}},{{1, 0},{0, 1}}}}

Arguments: afunction is a function whose variables are listed in alistOfV ars, where alistOfV ars should be of the form {x₁,x₂,…,x_n}.
The default options for NCConvexityRegion are:
  NCSimplifyDiagonal →False
  DiagonalSelection → False
  ReturnPermutation → False
  ReturnBorderVector → False
  AllPermutation → False
  NCSimplifyDiagonal is an option geared toward a similar option used in
NCLDUDecomposition. This will make sure that the pivots (or diagonal entries) are all first simplified with NCSimplifyRational before they are used to check that the pivots are all nonzero. Simplifying the pivots using NCSimplifyRational can be quite time consuming, so by default we commute everything and then use Mathematica simplification commmands. We do this only to convince ourselves that the pivot is nonzero. If all the pivots are zero using CommuteEverything we then revert to using NCSimplifyRational to verify our suspicions. Setting NCSimplifyDiagonal → True will bypass the commute everything step. (Note: Either way, the unsimplified form of the pivot is returned unless it is equal to zero.)
  The option AllPermutation tells NCConvexityRegion which of
NCLDUDecomposition or NCAllPermutationLDU to use. Setting AllPermutation to True will use
NCAllPermutationLDU,while False uses NCLDUDecomposition. The default value is AllPermutation → False. The following pertains to the case where AllPermutation is set to True. If you decide to do this, then you should also set DiagonalSelection to the permutations you would like NCAllPermutationLDU to use. Since different permutations return different diagonals, some diagonals are simpler to work with than others. On the other hand, if AllPermutation is set to False, which it defaults to, then NCConvexityRegion calls NCLDUDecomposition and what follows does not apply as no permutations are used. Different permutations return different diagonals. Some diagonals are simpler to work with than others. Because of this, we allow the user to select a sampling of different permutations. The total number of permutations will not be known until M is computed. After M is computed, the total number of possible permutations will be printed on the screen. DiagonalSelection → {n} returns the diagonals resulting from the first n permutations. DiagonalSelection→{k,n} returns the diagonals resulting from the k^th through n^th permutations. Since the total number of permutations is assumed to be unknown by the user, if n is too high, then n is replaced by the total number of permutations. Also, not all of the permutations are permissible. Because of this, NCLDUDecomposition automatically pivots if an invalid permutation is used for a particular step. This means it is possible that not all the diagonals returned result from different permutations. For this reason there is the option ReturnPermutation which if entered as True returns the permutations used for each resulting factorization. Finally, the user may wish to analyze the border vectors and may do so by setting ReturnBorderVector to True. This will cause NCConvexityRegion to return the border vectors v from the v^tMv factorization of the hessian. Now v^t will have the form So what will actually be returned is a list of the form This vector will be formed using a call to NCBorderVectorGather. Also, a call will be made to NCIndependenceCheck to determine, if possible, whether or not the elements of the above list are independent. The results of this check will be printed to the screen.

Comments / Limitations: None.

4.8.2 NCMatrixOfQuadratic[ , {H₁,…,H_n} ]

Aliases: None.

Description: NCMatrixOfQuadratic[ , {H₁,H₂,…,H_n} ] gives a vector matrix factorization of a symmetric quadratic function in noncommutative variables = {H₁,H₂,…,H_n} and their transposes.
NCMatrixOfQuadratic[ , {H₁,H₂,…,H_n} ], generates the list {left border vector, coefficient matrix, right border vector}. That is, Q is factored into the vector-matrix-vector product V []^T M

Q V[].ThevectorV[]islinearinandiscalledabordervectorofthequadraticfunctionQ.ThematrixM˙QiscalledthecoefficientmatrixofthequadraticfunctionQ.Arguments :EachtermofQisassumedtobeaquadraticexpressionintermsofthevariablesH˙1,H˙2,. . . ,H˙nandtheirtransposes(Qishomogeneous). Forexample,supposethatQ=3 tp[x]**y+3 tp[y]**xandthat = {x,y}.Then,NCMatrixOfQuadratic[Q,]gives{{{tp[x],tp[y]}},{{0, 3},{3, 0}},{{x},{y}}}.InMatrixForm,thislookslike* *. Ingeneral,supposeQisaquadraticfunctionoftwovariables, = {H,K},withalltransposeelementsHˆT, KˆToccuringbeforeallnon- transposeelements.ThenNCMatrixOfQuadraticwillreturntheleftbordervectorV[]ˆT,thematrixM˙Q,andtherightvectorV[]where M

Q := and V [ ⃗H] := ( | | | | | | | | ( HL1 1 HL1 2 ⋅⋅⋅ HL1 ℓ1 KL2 1 ⋅⋅⋅ KL2 ℓ2 ) | | | | | | | | ) forsomeL˙iˆj , i=1,. . . ,ℓj. The Lj i , i = 1,...,ℓj are called the coefficients of the border vector. The L1 i corresponding to H are distinct and only one may be the identity matrix (equivalently for the L2 i corresponding to K). The border vector V is the vector composed of H, K and Lj i. The matrix M is the matrix with Ai,j entries. Noncommutative quadratics which are not hereditary have a similar representation (which takes more space to write) for such a quadratic in H,K. For example, the border vector for a quadratic in H, HT , K, KT has the form V [H,K] = [ V1 V2 ] where we have V1 = ( (L1 1)T HT ,⋅⋅⋅ ,(L1 ℓ1)T HT ,(L2 1)T KT ,⋅⋅⋅ ,L(2 ℓ2)T KT ) and V2 = (~ L1 1H,⋅⋅⋅ , ~ L1 ℓ1H, ~ L2 1K,⋅⋅⋅ , ~ L2 ℓ2K) . We should emphasize that the size of the M representation of a noncommutative quadratic functions [H1,...,Hk] depends on the particular quadratic and not only on the number of arguments of the quadratic. There are noncommutative quadratic functions in one variable which have a representation with M a 102 × 102 matrix. The basic idea of NCMatrixOfQuadratic is that it searches for terms of form Left **X **Middle **Y **Right where X = Hi or HT i and Y = Hj or HT j for 1 ≤ (i,j) ≤ n. Terms of the form Left**X and Y **Right are used to form the left and right vectors. Each time the search finds a unique Right (Left) term causes the length of the right (left) border vector to be increased by one. The term Middle becomes the entries in the matrix M. Comments / Limitations: NCMatrixOfQuadratic will try to symmetrize the resulting matrix M. If NCMatrixOfQuadratic is unable to do this, an error message will be printed and { leftvector, matrix, rightvector } will be returned, where matrix is not symmetric and leftvector is not necessarily the transpose of rightvector. The vector-matrix-vector product should still be equal to the orginal quadratic expression.

4.8.3 NCIndependenceCheck[aListofLists,variable]

Aliases: None.

Description: NCIndependenceCheck[aListofLists,variable] is aimed at verifying whether or not a given set of polynomials are independent or not. It analyzes each list of polynomials in aListofLists separately. There are three possible types of outputs for each list in aListofLists. Two of them correspond to NCIndependenceCheck successfully determining whether or not the list of polynomials is independent. The third type of output corresponds to an unsuccessful attempt at determining dependence or independence. If a particular list is determined to be independent, True will be returned. If a list is determined to be dependent, a list beginning with False containing a set of coefficients which demonstrate independence will be returned. Finally, if NCIndependenceCheck cannot determine dependence or independence, it returns a list beginning with Undetermined containing other information which is illustrated below and described further in Comments/Limitations.

Arguments: aListofLists is a list containing a list of the polynomials which are suspected of being dependent. The argument variable will be subscripted and used to return the coefficient dependencies for each list. Below is an example of a list of four lists. The first two are dependent, the third is independent, and the fourth is undetermined.

Suppose you have four lists:

Then NCIndependenceCheck[List1,List2,List3,List4,λ] returns
{NewList1,NewList2,NewList3,NewList4} where:

In particular, what the above says is that List1.Newlist1[[2]] = 0,
and List2.Newlist2[[2]] = 0 (where “.” refers to the vector dot product). Therefore, the set of polynomials in List1 and List2 are dependent. List3 is independent. Note that List4 is clearly indpendent in the noncommutating case, and dependent in the commuting case. When such phenomena occur, NCIndependenceCheck is unable to determine whether or not the list of polynomials is independent. However, it does return to the user, a set of dependencies for the λ_i’s which must hold in order for the polynomials to sum to zero.

Comments / Limitations: IndependenceCheck first uses the CommuteEverything command to make the problem feasible. Therefore it is possible that polynomials are dependent if variables commute, and independent if not. So in this case, or when the the expression does not collapse to zero when using the commuting coefficients with the non commuting polynomials, then the list {Undertermined,expression,list} is returned. The list element expression is the sum of the polynomials with their corresponding λ’s. And finally, list yields a list of the dependencies for the coefficents.

4.8.4 NCBorderVectorGather[alist,varlist]

Aliases: None.

Description: NCBorderVectorGather[alist,varlist] can be used to gather the polynomial coefficents preceeding the elements given in varlist whenever they occur in alist. That is to say, alist is a vector with variable entries. Each entry should end with some term from varlist (or the transpose of some term from varlist). Then for each element of varlist the coefficients that appear in front of that element in alist are gathered together and placed inside a list. The list returned will be a list of lists, each entry a list of the coefficients corresponding to the respective entries in varlist and their transposes if they occur.

Arguments: The first argument alist is a list of polynomials, all of which end in terms from elements of the second argument, varlist, or in their transpose. alist need not be ordered in a particular way with respect to varlist. The preceeding is best explained in the following example.
Suppose List = Then NCBorderVectorGather[List,{k,h}] returns the following list Note that the vectors are gather in the pattern k,tp[k],h,tp[h]. This pattern will be the same despite the length of avarlist.

Comments / Limitations: None.

4.9 NCGuts

This section details the command NCGuts, which expands the meaning of “**”, tp[], and inv[].

4.9.1 NCStrongProduct1

Aliases: None.

Description: NCStrongProduct1 is an option of NCGuts. When True, ** serves to multiply matrices as well as maintaining its original function with noncommutative entries. This replaces the command MatMult. For example, is the same as In addition, tp and tpMat are the same. NCStrongProduct1 → False is the default.

Arguments: None.

Comments / Limitations: None.

4.9.2 NCStrongProduct2

Aliases: None.

Description: NCStrongProduct2 is an option of NCGuts. When set to true, if m is a matrix with noncommutative entries, inv[m] returns a formula expression for the inverse of m. The considerable limitations of NCInverse are still limitations in inv[m]. NCStrongProduct2 forces NCStrongProduct1. In other words, NCGuts[NCStrongProduct2-¿True] makes ”**” multiply matrices with noncommutative entries, just as NCGuts[NCStrongProduct1-¿True] does. NCStrongProduct2 → False is the default.

Arguments: None.

Comments / Limitations: None.

4.9.3 NCSetNC

Aliases: None.

Description: NCSetNC is an option of NCGuts. When set to false, all letters are automatically noncommutative unless SetCommutative makes them commutative. NCSetNC → False is the default.

Arguments: None.

Comments / Limitations: None.

4.10 Setting Properties of an element in an algebra

4.10.1 SetInv[a, b, c, …]

Aliases: None

Description: SetInv[a,b,c,…] sets all the symbols a, b, c, … to be invertible (i.e. invQ[a], invQ[b], invQ[c], … are set True).

Arguments: Symbols separated by commands

Comments / Limitations: If one does not set x to be invertible before the first use of invL[x] or invR[x], then NCAlgebra may not make the substitution from invL[x] **x to 1 or from x **invR[x] to 1 automatically.

4.10.2 SetSelfAdjoint[Symbols]

Aliases: None

Description: SetSelfAdjoint[a, b, …] will set a, b, … to be self-adjoint. The rules tp[a] := a, tp[b] :=b, … and aj[a] := a, aj[b] := b, … will be automatically applied. See SelfAdjointQ.

Arguments: Symbols is one or more symbols separated by commas.

Comments / Limitations: If one does not set x to be self adjoint before the first use of aj[x], then NCAlgebra may not make the substitution from aj[x] to x automatically. Similary for tp.

4.10.3 SelfAdjointQ[aSymbol]

Aliases: None

Description: SelfAdjointQ[x] will return True if SetSelfAdjoint[x] was executed previously. See SetSelfAdjoint.

Arguments: aSymbol is a symbol

Comments / Limitations: None

4.10.4 SetIsometry[Symbols]

Aliases: None

Description: SetIsometry[a,b,…] will set a, b, … to be isometries. If set the rules tp[a] ** a := Id, tp[b] ** b :=Id, … and aj[a] ** a := Id; aj[b] ** b := Id; … will be automatically applied. See IsometryQ.

Arguments: Symbols is one or more symbols separated by commas.

Comments / Limitations: If one does not set x to be an isometry before the first use of aj[x], then NCAlgebra may not make the substitution from aj[x] **x to 1 automatically. Similarly for tp.

4.10.5 IsometryQ[aSymbol]

Aliases: None

Description: IsometryQ[x] will return True if SetIsometry[x] was executed previously. See SetIsometry.

Arguments: aSymbol is a symbol.

Comments / Limitations: None

4.10.6 SetCoIsometry[Symbols]

Aliases: None

Description: SetCoIsometry[a,b,…] will set a, b, … to be co-isometries. The rules a ** tp[a] := Id, b ** tp[b] :=Id, … and a ** aj[a] := Id, b ** aj[b] := Id, … will be automatically applied. See CoIsometryQ.

Arguments: Symbols is one or more symbols separated by commas.

Comments / Limitations: If one does not set x to be a coisometry before the first use of aj[x], then NCAlgebra may not make the substitution from x **aj[x] to 1 automatically.

4.10.7 CoIsometryQ[aSymbol]

Aliases: None

Description: CoIsometryQ[x] will return True if SetCoIsometry[x] was executed previously. See SetCoIsometry.

Arguments: aSymbol is a symbol.

Comments / Limitations: None

4.10.8 SetUnitary[Symbols]

Aliases: None

Description: SetUnitary[a,b,…] will set a, b, … to be isometries and co-isometries. Also effects on UnitaryQ. See SetIsometry and SetCoIsometry.

Arguments: Symbols is one or more symbols separated by commas.

Comments / Limitations: If one does not set x to be a unitary before the first use of aj[x], then NCAlgebra may not make the substitution from x**aj[x] to 1 or from aj[x] **x to 1 automatically.

4.10.9 UnitaryQ[aSymbol]

Aliases: None

Description: UnitaryQ[x] will return True if SetUnitary[x] was executed previously. Caution: If one executes SetIsometry[x]; SetCoIsometry[x]; then x is unitary, but UnitaryQ remains uneffected. See SetUnitary.

Arguments: aSymbol is a symbol.

Comments / Limitations: None

4.10.10 SetProjection[Symbols]

Aliases: None

Description: SetProjection[a,b,…] will set a, b, … to be projections. The rules a ** a := a, b ** b :=b, … will be automatically applied. Caution: If one wants x to be a self-adjoint projection, then one must execute SetSelfAdjoint[x]; SetProjection[x]. See ProjectionQ.

Arguments: Symbols is one or more symbols separated by commas.

Comments / Limitations: If one does not set x to be a projection before the first use of x, then NCAlgebra may not make the substitution from x **x to x.

4.10.11 ProjectionQ[S]

Aliases: None

Description: ProjectionQ[x] will return true if SetProjection[x] was executed previously. See SetProjection.

Arguments: S is a symbol.

Comments / Limitations: None

4.10.12 SetSignature[Symbols]

Aliases: None

Description: When SetSignature[a] and SetSelfAdjoint[a] are executed, the rule a ** a := -1 will be automatically applied. See SetSelfAdjoint and SignatureQ.

Arguments: Symbols is one or more symbols separated by commas.

Comments / Limitations: If one does not set x to be a signature matrix and self adjoing before the first use of x, then NCAlgebra may not make the substitution from x **x to -1.

4.10.13 SignatureQ[Symbol]

Aliases: None

Description: SignatureQ[x] will return True if SetSignature[x] was executed previously. See SetSignature.

Arguments: Symbol is a symbol.

Comments / Limitations: None

4.11 Setting Properties of functions on an algebra

4.11.1 SetSesquilinear[Functions]

Aliases: SetSesq

Description: SetSesquilinear[a,b,c,…] sets a, b, c, … to be functions of two variables which are linear in the first variable and conjugate linear in the second variable. See SetBilinear.

Arguments: Functions is one or more symbols separated by commas.

Comments / Limitations: None

4.11.2 SesquilinearQ[aFunction]

Aliases: None

Description: SesquilinearQ[x] will return True if SetSesquilinear[x] was executed previously. See SetSesquilinear.

Arguments: aFunction is a symbol.

Comments / Limitations: None

4.11.3 SetBilinear[Functions]

Aliases: None

Description: SetBilinear[a,b,c,…] sets a, b, c, … to be functions of two variables which is linear in the first variable and linear in the second variable. See SetSesquilinear.

Arguments: Functions is one or more symbols separated by commas.

Comments / Limitations: None

4.11.4 BilinearQ[aFunction]

Aliases: None

Description: BilinearQ[x] will return True if SetBilinear[x] was executed previously. See SetBilinear.

Arguments: aFunction is a symbol.

Comments / Limitations: None

4.11.5 SetLinear[Functions]

Aliases: None

Description: SetLinear[b,c,d,…] sets b, c, d, … to be functions of one variable which are linear. See LinearQ.

Arguments: Functions is one or more symbols separated by commas.

Comments / Limitations: None

4.11.6 LinearQ[aFunction]

Aliases: None

Description: LinearQ[x] will return True if SetLinear[x] was executed previously. See SetLinear.

Arguments: aFunction is a symbol.

Comments / Limitations: None

4.11.7 SetConjugateLinear[Functions]

Aliases: None

Description: SetConjugateLinear[b,c,d,…] sets b, c, d, … to be functions of one variable which are conjugate linear. See ConjugateLinearQ.

Arguments: Functions is one or more symbols separated by commas.

Comments / Limitations: None

4.11.8 ConjugateLinearQ[aFunction]

Aliases: None

Description: ConjugateLinearQ[x] will return True if SetConjugateLinear[x] was executed previously. See SetConjugateLinear.

Arguments: aFunction is a symbol.

Comments / Limitations: None

4.11.9 SetIdempotent[Functions]

Aliases: None

Description: SetIdempotent[b,c,d,…] sets b, c, d, … to be functions of one variable such that, for example, b[b[z_]] := z; Common examples are inverse, transpose and adjoint. See IdempotentQ.

Arguments: Functions is one or more symbols separated by commas.

Comments / Limitations: None

4.11.10 IdempotentQ[aFunction]

Aliases: None

Description: IdempotentQ[x] will return True if SetIdempotent[x] was executed previously and False otherwise. See SetIdempotent.

Arguments: aFunction is a symbol.

Comments / Limitations: None

4.11.11 SetCommutingFunctions[ aFunction, anotherFunction]

Aliases: None

Description: SetCommutingFunctions takes exactly two parameters. SetCommutingFunctions[b, c] will implement the definitions b[c[z___]] := c[b[z]] /; Not[LeftQ[b,c]]; and c[b[z___]] := b[c[z]] /; LeftQ[b, c]; Common examples are the adjoint commuting with the transpose. Note: The above implementation will NOT lead to infinite loops. WARNING: If one says SetCommutingFunctions[b, c] and then sets only LeftQ[c,b], then neither of the above rules will be executed. Therefore, one must remember the order of the two parameters in the statement. One obvious helpful habit would be to use alphabetical order (i.e. say SetCommutingFunctions[aj, tp] and not the reverse). See CommutatingOperators and LeftQ.

Arguments: aFunction and anotherFunction are symbols.

Comments / Limitations: None

4.11.12 SetNonCommutativeMultiplyAntihomomorphism[ Functions]

Aliases: None

Description: SetNonCommutativeMultiplyAntihomomorphism[b,c,d,…] sets b, c, d, ... to be functions of one variable such that, for example, b[anything1**anything2] becomes b[anything2] **b[anything1] if ExpandQ[b] is True; b[anything2] ** b[anything1] becomes b[anything1 ** anything2] if ExpandQ[b] is False; Common examples are inverse, transpose and adjoint. NOTE: The synonym NCAntihomo is easier to type.

Arguments: Functions is one or more symbols separated by commas.

Comments / Limitations: None

4.12 Manipulating an Expression — less useful commands

4.12.1 NCTermArray[expr,aList,anArray]

Aliases: None

Description: NCTermArray[expr,aList,anArray] creates an array anArray whose contents represent the terms of expr sorted by degree. The variables anArray[”variables”], anArray[”types”] and elements such as anArray[x**x,y] and anArray[x **x **x,y **y] to hold the terms with 2 x‘s and 1 y and 3 x’s and 2 y’s, respectively (assuming that aList = {x,y}). You can reconstruct expr from anArray via ReconstuctTaylor[anArray].

Arguments: expr is an algebraic expression, aList is a list of variables and anArray is a symbol.

Comments / Limitations: Not available before NCAlgebra 1.0

The following is an example of the above command.

In[22]:= expr = x ** z ** x ** w + x ** z ** y ** w + z ** x ** x ** w +       z ** x ** y ** w;       In[23]:= NCTermArray[expr,{x,y},foo]     Out[23]= foo     In[24]:= ??foo   Global‘foo     foo["types"] = {{1, 1}, {2, 0}}     foo["variables"] := {x, y}     foo[x, y] = x ** z ** y ** w + z ** x ** y ** w     foo[x ** x, 1] = x ** z ** x ** w + z ** x ** x ** w     foo[x___] := 0

4.12.2 NCReconstructFromTermArray[anArray]

Aliases: None

Description: NCTermArray[expr,aList,anArray]; sets newexpr equal to expr.

Arguments: anArray is a symbol previously filled by NCTermArray

Comments / Limitations: Not available before NCAlgebra 1.0

If we continue with the example above we have the following

In[24]:= NCReconstructFromTermArray[foo]     Out[24]= x ** z ** x ** w + x ** z ** y ** w + z ** x ** x ** w +     >    z ** x ** y ** w

However, now one can also do some manipulation before reconstructing as shown below.

In[25]:= foo[x,y] = NCC[foo[x,y],y**w]   Out[25]= (x ** z + z ** x) ** y ** w     In[26]:= foo[x**x,1] = NCC[foo[x**x,1],x**w]     Out[26]= (x ** z + z ** x) ** x ** w     In[27]:= ??foo   Global‘foo     foo["types"] = {{1, 1}, {2, 0}}     foo["variables"] := {x, y}     foo[x, y] = NCC[x ** z ** y ** w + z ** x ** y ** w, y ** w]     foo[x ** x, 1] = NCC[x ** z ** x ** w + z ** x ** x ** w, x ** w]     foo[x___] := 0     In[27]:= NCReconstructFromTermArray[foo]   Out[27]= (x ** z + z ** x) ** x ** w + (x ** z + z ** x) ** y ** w

4.12.3 NCCompose[aVerySpecialList]

Aliases: NCCom

Description: NCCompose[NCDecompose[poly,a]] will reproduce poly. For example, NCCompose[NCDecompose[poly,a,b], 1, 0] will reconstruct the elements of poly which are of order 1 in a and of order 0 in b.

Arguments:

Comments / Limitations: Called within NCCollect. The average user would never use this.

4.12.4 NCDecompose[expr, listofsymbols]

Aliases: NCDec

Description: NCDecompose[poly, a] or NCDecompose[poly, a, b, c, ...] will produce a list of elements of poly in which elements of the same order of a (or the same order of a, b, c, ... ) are collected together.

Arguments:

Comments / Limitations: Called within NCCollect. The average user would never use this.

4.13 Utilities

Most of these utilities are for saving things. They probably do not work nor will you wish to use them in the Notebook environment.

4.13.1 SaveRules[expression, ’optional tag → ”message”’]

Aliases: SaveR

Description: Its main purpose is to control the Rules.temp file which records the rules used any time a Substitute command is used.

Arguments: SaveRules[On] turns on the Rules.temp for continuous recording of rules. SaveRules[Off] turns off the continuous record feature of Substitute commands, but any Substitute command can make a record in Rules.temp by using On as its optional argument. SaveRules[expression] will save the evaluated form of expression to the Rules.temp. SaveRules[expression,tag → ”message”] will save the evaluated form of ’expression’ to the Rules.temp file with a explanatory message.

Comments / Limitations: SaveRules[”ccc”], where ccc is a string, can be used to include comments into the Rules.temp file. SaveRules[mathematical expression] will record the mathematical expression without its definitions.

4.13.2 SaveRulesQ[]

Aliases: SaveRQ

Description: SaveRulesQ[] indicates the status of the continuous recording feature of the Substitute commands into the Rules.temp file by message and returns True if continuous records are being made and False if continuous records are not being made.

Arguments: None

Comments / Limitations: Messages can be suppressed or enabled by typing Off[ SaveRulesQ::Off] and On[SaveRulesQ::On].

4.13.3 FunctionOnRules[Rules, Function1, Function2, (optional On)]

Aliases: FORules

Description: It maps Function1 onto the left hand side and Function2 onto the right hand side of each rule in a set of rules, and returns the new set of rule. For example, FunctionOnRules[ { a→ x, b→ y }, Sin, Cos] gives {Sin[a] → Cos[x], Sin[b] → Cos[y] }

Arguments: Rules is a single rule or list of rules. Function1 and Function2 are any built-in Mathematica function, NCAlgebra function, pure function, or user-defined function.

Comments / Limitations: None

4.14 Deprecated Commands

The following commands are no longer supported in this version of NCAlgebra.

4.14.1 RandomMatrix[m,n,min,max,options]

Aliases: None.

Description: RandomMatrix[ m, n, min, max, options ] returns a random matrix of size m by n with entries between the values min and max.

Arguments: m and n are the number of rows and columns of the matrix. min and max are the minimum and maximum values of the entries in the matrix. The options are MatrixType and EntryType. The option MatrixType has values Any, Diagonal, or Symmetric. The default is MatrixType → Any which returns an ordinary unrestricted matrix. MatrixType → Diagonal returns random diagonal matrices. MatrixType → Symmetric returns random symmetric matrices. The option EntryType has values Integer or Real. The default is EntryType → Real which produces real floating point numbers as entries for the random matrix. EntryType → Integer produces integers as entries.

Comments / Limitations: Functionality provided by new RandomInteger, RandomDouble, etc, native Mathematica commands

4.14.2 CEEP

Aliases: None

Description: The “CEEP” file tells Mathematica that you want to record the functions you use during the session for later use or examination. It prompts you for a file name. Say you respond MYSESSION. CEEP records your session – two different ways into two different files – the first file (e.g. MYSESSION.m and MYSESSION.ex) records the In[] and Out[] lines of code you see on the screen and the second file (which contains the suffix .ex – MYSESSION.ex in the above example) saves just the commands which you type.

Functions stored in these files can be brought into a Mathematica session at a later time by typing ≪MYSESSION.ex and it executes. This file can also be modified in a text editor external to the Mathematica program. The “NCAlgebra.m” file contains the instructions to load NCAlgebra “packages” which allow the manipulation of noncommutative expressions.

Also, when using UNIX via a UNIX shell (rather than a Mathematica notebook), UNIX has a ’script’ utility which can be used. Type ’man script’ to find out more.

Arguments: None

Comments / Limitations: Fails inside a Notebook. You can essentially achieve the same functionality by using file=OpenAppend[”filename”]; AppendTo[$Echo, file]; If TEXformating is desired use file=OpenAppend[”filename”, FormatType - > NCTeXForm];

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