MATRICES AND MATRIX OPERATIONS: Unit 9

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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(9) Multiplication involving the unit matrix

OBJECTIVES :

• To define the unit matrix or identity matrix .

• To provide alternative methods of inputting the unit matrix.

• To provide an example of multiplication of a square matrix and the unit matrix.

• To provide an example of multiplication of the unit matrix and a column matrix .

• To provide an example of multiplication of a row matrix and the unit matrix.

• To specify and illustrate some properties of the unit matrix.

> restart : with(linalg, det, diag, multiply) :

A scalar matrix, in which all the diagonal elements are is called the unit matrix or identity matrix .

In Maple , there are two alternative methods of defining and inputting the unit matrix.

Consider a ( × ) unit matrix [ U ].

Method 1 . Using the diag function:

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

Method 2 . Using the identity indexing function:

> U := array(1..3, 1..3, identity) : U = matrix(U) ;

* * *

N.B. The unit matrix commutes in multiplication, i.e. for any square matrix [ A ]

[ U ] [ A ] = [ A ] [ U ] = [ A ]

where the order of [ U ] is the same as that of [ A ].

For example, consider a ( × ) unit matrix [ U ] and a ( × ) matrix [ A ] given as

> A := matrix(3, 3, [8, 2, 1, 1, 3, 0, 4, 2, 1]) : A = matrix(A) ;

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

> `UA` := multiply(U, A) : U*A = matrix(`UA`) ;

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

> matrix(U) * matrix(A) = matrix(`UA`) ;

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

> A := matrix(3, 3, [8, 2, 1, 1, 3, 0, 4, 2, 1]) : U := diag(1, 1, 1) :

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

> `AU` := multiply(A, U) : A*U = matrix(`AU`) ;

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

> matrix(A) * matrix(U) = matrix(`AU`) ;

* * *

N.B. The product of the unit matrix and a column matrix is the matrix unchanged .

For example, consider a ( × ) unit matrix [ U ] and a ( × ) column matrix [ CM ] given as

> CM := matrix(3, 1, [cm[11], cm[21], cm[31]]) : CM = matrix(CM) ;

The product [ U ] [ CM ] is the following ( × ) column matrix:

> `U CM` := multiply(U, CM) : U*CM = matrix(`U CM`) ;

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

> matrix(U) * matrix(CM) = matrix(`U CM`) ;

* * *

N.B. The product of a row matrix and the unit matrix is the matrix unchanged .

For example, consider a ( × ) row matrix [ RM ] given as

> RM := matrix(1, 3, [rm[11], rm[12], rm[13]]) : RM = matrix(RM) ;

and a ( × ) unit matrix [ U ].

The product [ RM ] [ U ] is the following ( × ) row matrix:

> `RM U` := multiply(RM, U) : RM*U = matrix(`RM U`) ;

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

> matrix(RM) * matrix(U) = matrix(`RM U`) ;

* * *

N.B. The determinant of a unit matrix is .

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

* * *

N.B. A unit matrix is not changed by transposition or inversion .

[ Refer to Unit (10) for the transpose of a matrix and to Unit (14) for the inverse of a matrix. ]

* * *

N.B. A unit matrix is an orthogonal matrix.

[ For the orthogonal matrix, refer to Unit (17). ]

* * *

Proceed to Unit (10) for " The transpose of a matrix ".

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