Kronecker Product with Mathematica

Kronecker Product.nb 1 Kronecker Product with Mathematica

Nicholas Wheeler Reed College Physics Department October 2000

Introduction

In several recent essays I have drawn extensively on properties of the Kronecker product—a concept not treated in most standard introductions to matrix theory. It is to open the door to experimentation in the area, and to describe the tools I used in some of my own exploratory work, that I offer the following material.

0. Matrices with doubly-indexed elements

The objects of interest to us are ordinary symbolic matrices. Mathematica has no objection to entries of the design

a11 a12 a13 a = J a21 a22 a23 N

a11, a12, a13 , a21, a22, a23 88 < 8 <<

% MatrixForm êê a11 a12 a13 J a21 a22 a23 N

Clear a @ D

But when we try to enter the subscripted design Kronecker Product.nb 2

a a a a = 11 12 13 J a21 a22 a23 N

— $RecursionLimit::reclim : Recursion depth of 256 exceeded.

— General::stop : Further output of $RecursionLimit::reclim will be suppressed during this calculation.

$Aborted

Mathematica complains. The curious fact—for which I cannot at present account—is that

p p p P = 11 12 13 J p21 p22 p23 N q q Q = 11 12 J q21 q22 N

p ,p ,p , p ,p ,p 88 11 12 13 < 8 21 22 23 <<

q ,q , q ,q 88 11 12 < 8 21 22 <<

work perfectly well: Mathematica appears to resolve the letters of the alphabet into two classes: some letters are "subscriptable," some aren't!

1. Mathematica's "Outer" command

The following command

K = Outer Times, P, Q @ D p q ,p q , p q ,p q , 8888 11 11 11 12 < 8 11 21 11 22 << p q ,p q , p q ,p q , 88 12 11 12 12 < 8 12 21 12 22 << p q ,p q , p q ,p q , 88 13 11 13 12 < 8 13 21 13 22 <<< p q ,p q , p q ,p q , 888 21 11 21 12 < 8 21 21 21 22 << p q ,p q , p q ,p q , 88 22 11 22 12 < 8 22 21 22 22 << p q ,p q , p q ,p q 88 23 11 23 12 < 8 23 21 23 22 <<<< Kronecker Product.nb 3

MatrixForm K @ D

p11 q11 p11 q12 p12 q11 p12 q12 p13 q11 p13 q12 ji zy j J p11 q21 p11 q22 NJp12 q21 p12 q22 NJp13 q21 p13 q22 N z j z j p q p q p q p q p q p q z j 21 11 21 12 22 11 22 12 23 11 23 12 z j z j J p21 q21 p21 q22 NJp22 q21 p22 q22 NJp23 q21 p23 q22 N z k {

captures well the essential meaning of the Kronecker product. But it yields a result which is not a matrix because not a list of lists (it is instead a list of lists of lists). And—since not a matrix—it cannot be manipulated like a matrix to establish properties of the Kronecker product, or to evaluate such Kronecker products as arise in particular calculations. We confront this problem: How to remove the internal parentheses?

Resolution of the problem requires familiarity with various list manipulation resources. Look at the following:

Flatten Part K, 1, 1, 1 , Part K, 1, 2, 1 , Part K, 1, 3, 1 @8 @ D @ D @ D

Flatten Part K, 1, 1, 1 , Part K, 1, 2, 1 , Part K, 1, 3, 1 , 8 @8 @ D @ D @ D

% MatrixForm êê

p11 q11 p11 q12 p12 q11 p12 q12 p13 q11 p13 q12 i y j p q p q p q p q p q p q z j 11 21 11 22 12 21 12 22 13 21 13 22 z j z j p21 q11 p21 q12 p22 q11 p22 q12 p23 q11 p23 q12 z j z j p21 q21 p21 q22 p22 q21 p22 q22 p23 q21 p23 q22 z k {

We have here removed the interior parentheses "by hand." The procedure works, but it would be tedious to have to execute all the steps each time we encounter a fresh Kronecker product. What we need are commands that serve to automate the procedure. Kronecker Product.nb 4

2. Automated construction of the Kronecker product

Recall, by way of preparation, that dimensions of an arbitrary matrix can be obtained by the commands illustrated below:

Dimensions P 1 * number of rows * @ DP TH L 2

Dimensions P 2 * number of columns * @ DP TH L 3

The double backets are entered Â[[Â and Â]]Â. Note also that "circled times" is produced by the keyboard strokes Âc*Â.

Much experimentation and many false starts have led me to the following composite command

Flatten @ Table Flatten Table Part K, i, j, k , j, Dimensions P 2 , @ @ @ @ D 8 @ DP T

p11 q11 p11 q12 p12 q11 p12 q12 p13 q11 p13 q12 ji zy j p11 q21 p11 q22 p12 q21 p12 q22 p13 q21 p13 q22 z j z j p q p q p q p q p q p q z j 21 11 21 12 22 11 22 12 23 11 23 12 z j p q p q p q p q p q p q z k 21 21 21 22 22 21 22 22 23 21 23 22 {

and thus to the following definition:

P_ ≈ Q_ := Flatten Table Flatten Table Part Outer Times, P, Q ,i,j,k , @ @ @ @ @ @ D D j, Dimensions P 2 , 8 @ DP T

We check it out in the generic case

P ≈ Q MatrixForm êê

and in a couple of cases the definition "has not seen before:" Kronecker Product.nb 5

a1 i y a = j a2 z j z j a3 z k { b1 ji zy j b2 z b = j z j b3 z j z j b4 z k { a1 , a2 , a3 88 < 8 < 8 <<

b1 , b2 , b3 , b4 88 < 8 < 8 < 8 <<

a ≈ b MatrixForm êê a1 b1 ji zy j a1 b2 z j z j a1 b3 z j z j z j a1 b4 z j z j a2 b1 z j z j z j a2 b2 z j z j z j a2 b3 z j z j a2 b4 z j z j z j a3 b1 z j z j a3 b2 z j z j z j a3 b3 z j z j a3 b4 z k {

r11 r12 r = J r21 r22 N s11 s12 s = J s21 s22 N

r11, r12 , r21, r22 88 < 8 <<

s11, s12 , s21, s22 88 < 8 <<

r ≈ s MatrixForm êê r11 s11 r11 s12 r12 s11 r12 s12 ji zy j r11 s21 r11 s22 r12 s21 r12 s22 z j z j r21 s11 r21 s12 r22 s11 r22 s12 z j z j r21 s21 r21 s22 r22 s21 r22 s22 z k {

Seems to work! Kronecker Product.nb 6

Some Kronecker product identities

At (3) in "Toy Quantum Field Theory: Populations of Indistinguishable Finite-State Systems" (Notes for a Reed College Physics Seminar, 1 Novembeer 2000) I list basic properties of the Kronecker product. Earlier versions of the list can be found on pages 32–33 of Classical Theory of Fields (1999) and at (63) in Chapter 1 of Advanced Quanatum Topics (2000), where I cite my ultimate sources (E. P. Wigner's Group Theory & its Application to the Quantum Theory of Atomic Spectra, P. Lancaster's Theory of Matrices and Richard Bellman's Introduc- tion to Matrix Analysis). Here I demonstrate those properties in illustrative concrete cases.

First we introduce some matrices to work with:

Clear P, Q @ D

p p p P = 11 12 13 J p21 p22 p23 N q q q Q = 11 12 13 J q21 q22 q23 N u u U = 11 12 J u21 u22 N v v V = 11 12 J v21 v22 N

w11 w12 w13 ji zy W = j w21 www w23 z j z j w31 w32 w33 z k { p ,p ,p , p ,p ,p 88 11 12 13 < 8 21 22 23 <<

q ,q ,q , q ,q ,q 88 11 12 13 < 8 21 22 23 <<

u ,u , u ,u 88 11 12 < 8 21 22 <<

v ,v , v ,v 88 11 12 < 8 21 22 <<

w ,w ,w , w ,w ,w , w ,w ,w 88 11 12 13 < 8 21 ww 23 < 8 31 32 33 <<

ü Scalar associativity:

6 P ≈ Q ã 6 P ≈ Q H L H L True Kronecker Product.nb 7

6 P ≈ Q ã P ≈ 6 Q H L H L True

ü Distributivity:

Simplify P + Q ≈ U ã P ≈ U + Q ≈ U @H L D True

ü Kronecker associativity:

Simplify P ≈ U ≈ W ã P ≈ U ≈ W @H L H LD True

ü Transposition rule:

Simplify Transpose P ≈ U ã Transpose P ≈ Transpose U @ @ D @ D @ DD True

ü Trace rule:

Simplify Tr P ≈ U ã Tr P Tr U @ @ D @ D @ DD True

NOTE that Mathematica assigns a special, non-matrix-theoretic meaning to the word "trace."

ü Determinantal rule (both matrices square, but not necessarily co-dimensional):

Dimensions U @ D Dimensions W @ D 2, 2 8 <

3, 3 8 <

Simplify Det U ≈ W ã Det U 3 Det W 2 @ @ D @ D @ D D True

NOTE that each factor on the right wears the other's dimension as an exponent. Kronecker Product.nb 8

ü Inversion rule:

Simplify Inverse U ≈ W ã Inverse U ≈ Inverse W @ @ D @ D @ DD True

NOTE: That is a fairly amazing property, and took Mathematica several seconds to verify.

ü The amazing criss-cross rule:

Define a pair of additional matrices:

x1 ji zy j x2 z j z X = j x3 z j z j z j x4 z j z j x5 z k {

y11 y12 ji zy Y = j y21 y22 z j z j y31 y32 z k { Z = z z H 1 2 L x , x , x , x , x 88 1 < 8 2 < 8 3 < 8 4 < 8 5 <<

y ,y , y ,y , y ,y 88 11 12 < 8 21 22 < 8 31 32 <<

z ,z 88 1 2 <<

Dimensions P @ D Dimensions X @ D Dimensions Y @ D Dimensions Z @ D 2, 3 8 <

5, 1 8 <

3, 2 8 <

1, 2 8 <

The point is that we have now in hand a quartet of matrices for which all of the ordinary matrix products encountered in the following identity are meaningful. Kronecker Product.nb 9

Simplify P ≈ X . Y ≈ Z ã P.Y ≈ X.Z @H L H L H L H LD True

The matrix in question is a 10 x 4 mess, which I do not write down only because it overruns the right margin.

NOTE: Lancaster discusses this identity only in the case in which P and Y are co-dimension- ally square (m x m, let us say), and so are X and Z, though the latter may be of some different square dimension (n x n). He shows the identity to have a number of interesting corollaries. But that it holds under much weaker conditions (any conditions sufficient to insure multiplica- tive conformation) is my own little discovery, and central to the work cited in my introduction.