MATRICES AND MATRIX OPERATIONS: Unit 10

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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(10) The transpose of a matrix

OBJECTIVES :

• To define the transpose of a matrix and provide examples of transposition operation.

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

• To specify and illustrate general properties of transposition operation.

• To stress and show that transposition of a vector with Maple does not yield a column matrix .

• To introduce the function equal for testing if matrices are equal.

> restart : with(linalg, diag, equal, multiply, transpose) :

The transpose of a matrix ( or transposed matrix ) is the matrix resulting from interchanging the rows and columns in the given matrix. If the given matrix is of order ( × ) , then its transpose is the matrix of order ( × ) .

Consider a ( × ) matrix [ A ] given as

> A := matrix(3, 2, [ a[11], a[12], a[21], a[22], a[31], a[32] ]) : A = matrix(A) ;

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

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

This operation may be displayed in "like-in-a-book" form, viz.

> Transp(matrix(A)) = matrix(`transp(A)`) ;

* * *

N.B. The transpose of a diagonal matrix is the same as the matrix itself.

Exemplarily, the transpose of a ( × ) diagonal matrix [ B ] given as

> B := diag(3, 0, 2, 5) : B = matrix(B) ;

is the following ( × ) diagonal matrix:

> `transp(B)` := transpose(B) : Transp(B) = matrix(`transp(B)`) ;

This operation may be displayed in "like-in-a-book" form, viz.

> Transp(matrix(B)) = matrix(`transp(B)`) ;

* * *

N.B. Transposition of a square matrix does not change the determinant of the matrix.

[ For the determinant of a matrix, refer to Unit (11). ]

* * *

N.B. Transposition does not change the s ymmetric matrix.

Exemplarily, the transpose of a ( × ) symmetric matrix [ C ] given as

> C := matrix(3, 3, [1, a, a^2, a, a^2, 1, a^2, 1, a]) : C = matrix(C) ;

is the following ( × ) symmetric matrix:

> `transp(C)` := transpose(C) : Transp(C) = matrix(`transp(C)`) ;

This operation may be displayed in "like-in-a-book" form, viz.

> Transp(matrix(C)) = matrix(`transp(C)`) ;

* * *

N.B. Transposition changes sign of a skew-symmetric matrix but the matrix is otherwise unchanged.

Exemplarily, the transpose of a ( × ) skew-symmetric matrix [ E ] given as

> E := matrix(3, 3, [0, a, b, -a, 0, c, -b, -c, 0]) : E = matrix(E) ;

is the following ( × ) skew-symmetric matrix:

> `transp(E)` := transpose(E) : Transp(E) = matrix(`transp(E)`) ;

This operation may be displayed in "like-in-a-book" form, viz.

> Transp(matrix(E)) = matrix(`transp(E)`) ;

* * *

N.B. According to the definition of the transpose of a matrix, the transpose of a ( × ) row matrix is an ( × ) column matrix , in which .

Exemplarily, the transpose of a ( × ) row matrix [ RM ] given as

> RM := matrix(1, 3, [a, b, c]) : RM = matrix(RM) ;

is the following ( × ) column matrix:

> `transp(RM)` := transpose(RM) : Transp(RM) = matrix(`transp(RM)`) ;

This operation may be displayed in "like-in-a-book" form, viz.

> Transp(matrix(RM)) = matrix(`transp(RM)`) ;

* * *

N.B. In some textbooks, the following 'shortcut notation' is used in relation to a vector

to mean "convert vector to a column matrix by transposition".

Let us analyse this operation in Maple .

If the vector is assigned the name RV, viz.

> RV := array([a, b, c]) : RV = eval(RV) ;

then, transposition operation of [ RV ] yields

> `transp(RV)` := transpose(RV) : Transp(RV) = transpose(eval(RV)) ;

In executing this operation, Maple treats vector [ RV ] as if it were a column vector and, consequently, transposition of [ RV ] yields a row vector . This is evidenced by the following results.

(a) The product [ RV ] (Transp [ RV ] ), which is a ( × ) symmetric matrix:

> `RV transp(RV)` := evalm(RV &* `transp(RV)`) : RV * Transp(RV) = eval(`RV transp(RV)`) ;

This corresponds to the result of multiplication of a column matrix and a row matrix – refer to Section B of Unit (5).

(b) The product (Transp [ RV ] ) [ RV ] , which is a scalar (number):

> `(transp(RV)) RV` := evalm(`transp(RV)` &* RV) : `Transp(RV)` * RV = eval(`(transp(RV)) RV`) ;

This corresponds to the result of multiplication of a row matrix and a column matrix – refer to Section A of Unit (5).

Thus, transposition of a (row) vector does not yield a column matrix in Maple .

* * *

N.B. According to the definition of the transpose of a matrix, the transpose of an ( × ) column matrix is a ( × ) row matrix .

Exemplarily, the transpose of a ( × ) column matrix [ CM ] given as

> CM := matrix(3, 1, [d, e, f]) : CM = matrix(CM) ;

is the following ( × ) row matrix:

> `transp(CM)` := transpose(CM) : Transp(CM) = matrix(`transp(CM)`) ;

This operation may be displayed in "like-in-a-book" form, viz.

> Transp(matrix(CM)) = matrix(`transp(CM)`) ;

* * *

N.B. The transpose of the product of a matrix [ A ] and a scalar equals the matrix transpose multiplied by the scalar

Transp( k [ A ] ) = k Transp [ A ]

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

> A := matrix(2, 3, [ a[11], a[12], a[13], a[21], a[22], a[23] ]) : A = matrix(A) ;

(a) The transpose of the product of the matrix [ A ] and scalar , Transp( k [ A ] ) , is the following (3 × 2) matrix:

> `transp(kA)` := transpose(k*A) : Transp(k*A) = matrix(`transp(kA)`) ;

(b) The product of the matrix transpose multiplied by the scalar, k Transp [ A ], is the following (3 × 2) matrix:

> `k transp(A)` := evalm(k * transpose(A)) : k * Transp(A) = matrix(`k transp(A)`) ;

The equal function applied to the resultant matrices of (a) and (b), i.e.

> equal(`transp(kA)`, `k transp(A)`) ;

returns the Boolean value , which verifies that both matrices are equal.

* * *

N.B. The product of the transpose of a square matrix [ A ] and the matrix is a symmetric matrix. The product of the same square matrix [ A ] and its transpose is also a symmetric matrix but both resultant matrices are different.

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

> A := matrix(2, 2, [a, b, c, d]) : A = matrix(A) ;

(a) The product (Transp [ A ] ) [ A ] is the following ( × ) symmetric matrix:

> `(transp(A)) A` := evalm(transpose(A) &* A) : Transp(A) * A = matrix(`(transp(A)) A`) ;

(b) The product [ A ] (Transp [ A ] ) is the following ( × ) symmetric matrix:

> `A transp(A)` := evalm(A &* transpose(A)) : A * `Transp(A)` = matrix(`A transp(A)`) ;

The matrices of (a) and (b) are symmetric but different.

* * *

N.B. The sum of a square matrix and its transpose is a symmetric matrix.

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

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

The sum [ A ] + Transp [ A ] is the following ( × ) symmetric matrix:

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

This operation may be displayed in "like-in-a-book" form, viz.

> matrix(A) + Transp(matrix(A)) = matrix(`A+transp(A)`) ;

* * *

N.B. The difference of a square matrix and its transpose is a skew-symmetric matrix.

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

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

The difference [ A ] – Transp [ A ] is the following ( × ) skew-symmetric matrix:

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

This operation may be displayed in "like-in-a-book" form, viz.

> matrix(A) - Transp(matrix(A)) = matrix(`A-transp(A)`) ;

* * *

N.B. It follows immediately from the two preceding properties that any square matrix satisfies the formula

[ A ] = ( [ A ] + Transp [ A ] ) + ( [ A ] – Transp [ A ] )

where

• term ½ ( [ A ] + Transp [ A ] ) is referred to as the symmetric part of square matrix [ A ],

• term ½ ( [ A ] – Transp [ A ] ) is referred to as the antisymmetric part of square matrix [ A ].

This formula implies that a square matrix can be expressed as the sum of symmetric and antisymmetric parts of the matrix. The following example supports and illustrates these names.

Consider a ( × ) matrix [ A ] given as

> A := matrix(3, 3, [a, b, c, d, e, f, g, h, i]) : A = matrix(A) ;

(a) Compute the symmetric part of [ A ] as ½ ( [ A ] + Transp [ A ] ) :

> sym(A) :=evalm(1/2*(A + transpose(A))) : Sym(A) = matrix(sym(A)) ;

(b) Compute the antisymmetric part of [ A ] as ½ ( [ A ] – Transp [ A ] ) :

> antisym(A) := evalm(1/2*(A - transpose(A))) : Antisym(A) = matrix(antisym(A)) ;

(c) The sum of the two parts of [ A ] yields the matrix

> `sym(A)+antisym(A)` := evalm(sym(A) + antisym(A)) :

> Sym(A) + Antisym(A) = matrix(`sym(A)+antisym(A)`) ;

which is the original square matrix [ A ].

* * *

N.B. Transposition is reflexive , i.e. the transpose of a matrix transpose is the matrix itself

Transp(Transp [ A ] ) = [ A ]

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

> A := matrix(3, 2, [ a[11], a[12], a[21], a[22], a[31], a[32] ]) : A = matrix(A) ;

The transpose of the transpose of the matrix, Transp(Transp [ A ] ) , is the following ( × ) matrix:

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

This operation may be displayed in "like-in-a-book" form, viz.

> Transp(Transp(matrix(A))) = matrix(`transp(transp(A))`) ;

* * *

N.B. The transpose of the sum of two matrices is the sum of the two transposed matrices

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

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

> A := matrix(2, 3, [a[11], a[12], a[13], a[21], a[22], a[23]]) :

> B := matrix(2, 3, [b[11], b[12], b[13], b[21], b[22], b[23]]) :

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

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

> `transp(A+B)` := transpose(evalm(A + B)) : Transp(A + B) = matrix(`transp(A+B)`) ;

(b) The sum of the two transposed matrices, Transp [ A ] + Transp [ B ], is the following ( × ) matrix:

> `transp(A)+transp(B)` := evalm(transpose(A) + transpose(B)) :

> Transp(A) + Transp(B) = matrix(`transp(A)+transp(B)`) ;

This operation may be displayed in "like-in-a-book" form, viz.

> Transp(matrix(A) + matrix(B)) = matrix(`transp(A)+transp(B)`) ;

* * *

N.B. The transpose of the product of two matrices is the product of the two transposed matrices but in the reverse order

Transp( [ A ] [ B ] ) = Transp [ B ] Transp [ A ]

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

> A := matrix(2, 3, [a[11], a[12], a[13], a[21], a[22], a[23]]) :

> B := matrix(3, 2, [b[11], b[12], b[21], b[22], b[31], b[32]]) :

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

(a) The transpose of the product of the two matrices, Transp( [ A ] [ B ] ) , is the following ( × ) matrix:

> `transp(AB)` := transpose(multiply(A, B)) : Transp(A * B) = matrix(`transp(AB)`) ;

(b) The product of the two transposed matrices multiplied in the reverse order, Transp [ B ] Transp [ A ], is the following ( × ) matrix:

> `transp(B) transp(A)` := multiply(transpose(B), transpose(A)) :

> Transp(B) * Transp(A) = matrix(`transp(B) transp(A)`) ;

This operation may be displayed in "like-in-a-book" form, viz.

> Transp(matrix(A)*matrix(B)) = matrix(`transp(B) transp(A)`) ;

* * *

Proceed to Unit (11) for " The determinant of a matrix ".

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