MATRICES AND MATRIX OPERATIONS: Unit 15

Dr. Wlodzislaw Kostecki

The Papua New Guinea University of Technology (PNGUT)

Department of Electrical and Communication Engineering

Lae, Morobe Province

Papua New Guinea

Copyright © 2000 by Wlodzislaw Kostecki

All rights reserved

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(15) Integer exponentiation of matrices

OBJECTIVES :

• To define the operation of integer exponentiation of square matrices.

• To provide alternative methods of exponentiation operation with Maple .

• To provide definitions and examples of nilpotent and idempotent matrices and illustrate their properties.

• To show the effect of exponentiation operation on specific types of matrices.

• To specify and illustrate some properties of exponentiation operation.

• To stress and show that a square matrix raised to the power yields in Maple the scalar value , and not a zero matrix.

> restart : with(linalg, definite, det, diag, eigenvals, inverse, multiply, transpose) :

Positive- and negative-integer exponentiation (integer power) of a matrix is defined only for square matrices. Integer exponentiation of a matrix is multiple multiplication of the matrix with itself. Therefore, to satisfy the multiplication conformability rule , this operation requires that a matrix must be square.

A. Positive-integer power of a matrix

The positive-integer power of a square matrix of order ( × ) or ( × ) is a matrix of the same order. The resultant is a product matrix that is obtained by multiple multiplication of a given matrix with itself. For example, the square of the matrix [ A ] is defined by the relation

[ A ] ^ = [ A ] [ A ]

Exemplarily, consider a ( × ) matrix [ A ] given as

> A := matrix(2, 2, [a[11], a[12], a[21], a[22]]) : A = matrix(A) ;

The square of the matrix, [ A ] ^ , may be obtained using either of the following alternative methods.

Method 1 . Using the evalm function:

> `A^2` := evalm(A^2) : A^2 = matrix(`A^2`) ;

Method 2 . Using the multiply function:

> `A^2` := multiply(A, A) : A^2 = matrix(`A^2`) ;

As a numerical example, compute the square of a ( × ) matrix [ A ] given as

> A := matrix(2, 2, [1, 2, 2, -4]) : A = matrix(A) ;

The square of [ A ] is the following ( × ) matrix:

> `A^2` := evalm(A^2) : A^2 = matrix(`A^2`) ;

or

> `A^2` := multiply(A, A) : A^2 = matrix(`A^2`) ;

* * *

N.B. The above matrix [ A ] is a unique square root of matrix [ A ] ^ since the latter is a positive definite matrix as verified by the Boolean value returned by the function definite , viz.

> definite(`A^2`, 'positive_def') ;

[ For the function square root of a matrix, refer to Section B of Unit (23). ]

* * *

N.B. If a square matrix vanishes upon being raised to some positive power , then the matrix is said to be nilpotent of index . This implies that raising such a matrix to any integer power greater than the index will also result in a matrix.

Exemplarily, consider a ( × ) matrix [ A ] given as

> A := matrix(3, 3, [1, 5, -2, 1, 2, -1, 3, 6, -3]) : A = matrix(A) ;

and the index .

The matrix [ A ] raised to this power yields the ( × ) zero matrix

> `A^3` := evalm(A^3) : A^3 = matrix(`A^3`) ;

or, the matrix [ A ] has vanished upon raising it to power .

Notice that the determinant of a nilpotent matrix is :

> `det(A)` := det(A) : Det(A) = `det(A)` ;

Notice that eigenvalues of a nilpotent matrix are all :

> charroots(A) := eigenvals(A) : char_roots(A) = charroots(A) ;

[ For eigenvalues of matrices, refer to Unit (21). ]

* * *

N.B. If a square matrix is unchanged under multiplication by itself, then the matrix is said to be idempotent . This implies that raising such a matrix to any integer power greater than will also not change the matrix.

Exemplarily, consider a ( × ) matrix [ A ] given as

> A := matrix(3, 3, [2, -2, -4, -1, 3, 4, 1, -2, -3]) : A = matrix(A) ;

The square of [ A ] is the same matrix

> `A^2` := evalm(A^2) : A^2 = matrix(`A^2`) ;

Notice that the determinant of an idempotent matrix is :

> `det(A)` := det(A) : Det(A) = `det(A)` ;

* * *

N.B. If a diagonal matrix is raised to some positive power , then its elements are raised to the same power.

Exemplarily, consider a ( × ) diagonal matrix [ A ] given as

> A := diag(a[11], 0, a[33], a[44]) : A = matrix(A) ;

and the power .

The matrix [ A ] raised to this power yields the following diagonal matrix:

> `A^5` := evalm(A^5) : A^5 = matrix(`A^5`) ;

* * *

N.B. The transpose of a square matrix raised to some (positive or negative) power is equal to the transpose raised to this power

Transp( [ A ] ^ p ) = (Transp [ A ] )^ p

Exemplarily, consider a ( × ) matrix [ A ] given as

> A := matrix(3, 3, [2, -1, 4, 1, 3, 3, -1, 2, 0]) : A = matrix(A) ;

and the power .

(a) The transpose of the matrix raised to this power, Transp( [ A ] ^ ) , is the following ( × ) matrix:

> `transp(A^3)` := transpose(evalm(A^3)) : Transp(A^3) = matrix(`transp(A^3)`) ;

(b) The transpose raised to this power, (Transp [ A ] )^ , is the following ( × ) matrix:

> `(transp(A))^3` := evalm((transpose(A))^3) : [Transp(A)]^3 = matrix(`(transp(A))^3`) ;

Both resultant matrices are equal.

* * *

N.B. If two square matrices, [ A ] and [ B ], are commuting matrices, then and only then

( [ A ] + [ B ]) ^ = [ A ] ^ + [ A ][ B ] + [ B ] ^

otherwise

( [ A ] + [ B ]) ^ = [ A ] ^ + [ A ][ B ] + [ B ][ A ] + [ B ] ^

Exemplarily, consider two ( × ) commuting matrices [ A ] and [ B ] given as

> A := matrix(2, 2, [3, 4, 2, 3]) : B := matrix(2, 2, [3, -4, -2, 3]) : A = matrix(A) ; B = matrix(B) ;

(a) The sum of the two matrices squared, ( [ A ] + [ B ] )^ , is the following ( × ) scalar matrix:

> `(A+B)^2` := evalm((A+B)^2) : (A + B)^`2` = matrix(`(A+B)^2`) ;

(b) The sum [ A ] ^ + [ A ][ B ] + [ B ] ^ is the following ( × ) scalar matrix:

> `A^2+2AB+B^2` := evalm(A^2 + 2*A &* B + B^2) : A^2+2*A*B+B^2 = matrix(`A^2+2AB+B^2`) ;

The resultant matrices of (a) and (b) are equal.

* * *

N.B. In general, the integer power of the product of square matrices [ A ] and [ B ] of the same order is not equal to the product of either matrix raised to the same power

( [ A ] [ B ] )^n is not equal to [ A ] ^n [ B ] ^n

unless the matrices are commuting matrices ( for which [ A ] [ B ] = [ B ] [ A ] ) or diagonal matrices.

(1) Exemplarily, let the natural number and ( × ) matrices [ A ] and [ B ] be given as

> A := matrix(3, 3, [2, 3, 1, 2, -2, -2, -1, 2, 1]) : B := matrix(3, 3, [3, 1, -3, 0, 2, 6, -7, 1, 2]) :

> A = matrix(A) ; B = matrix(B) ;

(a) The square of the product of both matrices, ( [ A ] [ B ] )^2 , is the following matrix:

> `(AB)^2` := evalm((A &* B)^2) : (A*B)^`2` = matrix(`(AB)^2`) ;

(b) The product of either matrix raised to the same power, [ A ] ^2 [ B ] ^2 , is the following matrix:

> `A^2 B^2` := evalm(A^2 &* B^2) : A^2 * B^2 = matrix(`A^2 B^2`) ;

The resultant matrices of (a) and (b) are different.

(2) Let and ( × ) commuting matrices [ A ] and [ B ] be the same that are used in Unit (4), i.e.

> A := matrix(2, 2, [6, 8, 4, 6]) : B := matrix(2, 2, [15, 20, 10, 15]) : A = matrix(A) ; B = matrix(B) ;

(a) The square of the product of both matrices, ( [ A ] [ B ] )^2 , is the following matrix:

> `(AB)^2` := evalm((A &* B)^2) : (A*B)^`2` = matrix(`(AB)^2`) ;

(b) The product of either matrix raised to the same power, [ A ] ^2 [ B ] ^2 , is the following matrix:

> `A^2 B^2` := evalm(A^2 &* B^2) : A^2 * B^2 = matrix(`A^2 B^2`) ;

The resultant matrices of (a) and (b) are equal.

(3) Let and ( × ) diagonal matrices [ A ] and [ B ] be given as

> A := diag(3, -5, 7) : B := diag(-2, 4, 6) : A =matrix(A) ; B = matrix(B) ;

(a) The square of the product of both matrices, ( [ A ] [ B ] )^2 , is the following matrix:

> `(AB)^2` := evalm((A &* B)^2) : (A*B)^`2` = matrix(`(AB)^2`) ;

(b) The product of either matrix raised to the same power, [ A ] ^2 [ B ] ^2 , is the following matrix:

> `A^2 B^2` := evalm(A^2 &* B^2) : A^2 * B^2 = matrix(`A^2 B^2`) ;

The resultant matrices of (a) and (b) are equal.

* * *

N.B. The zero matrix raised to any positive integer power is the matrix unchanged.

As an example, consider the ( × ) matrix [ 0 ]

> `0` := matrix(2, 2, [0, 0, 0, 0]) : `0` = matrix(`0`) ;

The square of matrix [ 0 ] is the following matrix:

> `0^2` := evalm(`0`^2) : `0`^2 = matrix(`0^2`) ;

* * *

N.B. Literature sources use a convention that a square matrix raised to the power is equal to the appropriately sized unit matrix , i.e.

[ A ] ^ = [ U ]

In Maple , the result of this exponentiation is the scalar value .

Exemplarily, consider a ( × ) matrix [ A ] given as

> A := matrix(2, 2, [a[11], a[12], a[21], a[22]]) : A = matrix(A) ;

Raising matrix [ A ] to the power yields

> `A^0` := evalm(A^0) : A ^ `0` = `A^0` ;

* * *

B. Negative-integer power of a matrix

The negative-integer power of a square matrix of order ( × ) or ( × ) is a matrix of the same order. The resultant is a product matrix that is obtained by multiple multiplication of the inverse of a given matrix. For example, the negative second power of a matrix [ A ] is defined by the relation

[ A ] ^( ) = (Inv [ A ] )^

Exemplarily, consider a ( × ) matrix [ A ] given as

> A := matrix(2, 2, [a[11], a[12], a[21], a[22]]) : A = matrix(A) ;

The negative second power of [ A ] may be obtained using either of the following alternative methods.

Method 1 . Using the evalm function:

> `A^(-2)` := evalm(A^(-2)) : A ^ ` -2` = matrix(`A^(-2)`) ;

or, in a more compact form,

> `A^(-2)` := map(x->normal(x, expanded), `A^(-2)`) : A ^ ` -2` = matrix(`A^(-2)`) ;

Method 2 . Using the evalm and inverse functions:

> `A^(-2)` := evalm((inverse(A))^2) : A ^ ` -2` = matrix(`A^(-2)`) ;

or, in a compact form,

> `A^(-2)` := map(x->normal(x, expanded), `A^(-2)`) : A ^ ` -2` = matrix(`A^(-2)`) ;

As a numerical example, compute the negative second power of a ( × ) matrix [ A ] given by

> A := matrix(2, 2, [-2, 4, 6, 1]) : A = matrix(A) ;

The negative second power of [ A ] is the following ( × ) matrix:

> `A^(-2)` := evalm(A^(-2)) : A ^ ` -2` = matrix(`A^(-2)`) ;

or

> `A^(-2)` := evalm((inverse(A))^2) : A ^ ` -2` = matrix(`A^(-2)`) ;

Floating-point evaluation of the negative second power of the matrix [ A ] gives the approximation

> `A^(-2)` := evalf(evalm(`A^(-2)`)) : A ^ ` -2` = matrix(`A^(-2)`) ;

* * *

N.B. The unit matrix raised to any (positive or negative) integer power is the matrix unchanged.

Exemplarily, compute the square and power of the ( × ) unit matrix [ U ]

> U := diag(1, 1) : U = matrix(U) ;

> `U^2` := evalm(U^2) : `U^(-3)` := evalm(U^(-3)) :

> U^2 = matrix(`U^2`) ; U ^ ` -3` = matrix(`U^(-3)`) ;

* * *

Proceed to Unit (16) for " The complex matrix and its conjugate ".

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