MATRICES AND MATRIX OPERATIONS: Unit 29

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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(29) Integration of matrices comprising functions

OBJECTIVES :

• To state the condition necessary for a matrix containing functions of one variable to have an integral with respect to this variable.

• To introduce the concept of an integrable matrix.

• To provide a definition of the indefinite integral of a matrix in the form of a symbolic expression.

• To provide an example of an indefinite integral of a rectangular matrix whose elements are functions.

• To specify and illustrate properties of matrix integration.

• To provide a definition of the definite integral of a matrix in the form of a symbolic expression.

• To state the condition necessary for a matrix containing functions of one variable to have a proper integral with respect to this variable.

• To provide an example of a proper integral of a square matrix whose elements are functions.

• To provide alternative methods of exact and floating-point evaluation of a proper integral of a matrix.

• To show how to evaluate a proper integral of a matrix whose elements are functions but the integration limits have a symbolic form and their numerical values become known in a later phase of computation.

• To point out that Maple can compute improper integrals of matrices comprising functions.

• To define the improper integral of a matrix and specify the two distinct cases of such an integral.

• To provide two examples of computation of improper matrix integrals due to infinite interval of integration.

• To provide two examples of computation of improper matrix integrals due to discontinuity of the integrand in the interval of integration.

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

If some or all elements of a matrix [ A ] are functions of one variable t , it is convenient to denote the matrix as [ A ( ) ] when considering integration of such a matrix.

A. Indefinite integrals

When those elements of a matrix [ A ( ) ] that are functions are all differentiable with respect to in some common interval < < , then an indefinite integral of [ A ( ) ] with respect to may be defined. Then, also the matrix [ A ( ) ] is said to be integrable in the interval < < .

Consider a ( × ) matrix [ A ( ) ] given as

> A(t) := matrix(2, 3, [a[11](t), a[12](t), a[13](t), a[21](t), a[22](t), a[23](t)]) : 'A(t)' = A(t) ;

The indefinite integral of [ A ( ) ] with respect to is defined to be the matrix of the same order with elements , viz.

> `indef_int(A)` := map(Int, A(t), t) : Int('A(t)', t) = matrix(`indef_int(A)`) ;

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

> Int(A(t), t) = matrix(`indef_int(A)`) ;

* * *

N.B. The integration of the matrix elements returns the most general antiderivative or primitive of each element with the tacit understanding that all constants of integration are .

* * *

For example, let the elements of a ( × ) matrix [ A ( ) ] be the functions as shown

> A(t) := matrix(2, 3, [2*exp(-3*t), sin(t/2), cosh(2*t), 3/sqrt(t), 2*t^2, 3*ln(2*t)]) : 'A(t)' = A(t) ;

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

> `indef_int(A)` := map(int, A(t), t) : Int('A(t)', t) = matrix(`indef_int(A)`) ;

or, in "like-in-a-book" form,

> Int(A(t), t) = matrix(`indef_int(A)`) ;

* * *

N.B. If two matrices, [ A ( ) ] and [ B ( ) ], are conformable for addition and integrable in some common interval, the indefinite integral of their sum is equal to the sum of the antiderivatives of both matrices

For example, let ( × ) matrices [ A ( ) ] and [ B ( ) ] be given as

> A(t) := matrix(2, 3, [t, t^3, -1, t^2, 2*t, -3*t^2]) : B(t) := matrix(2, 3, [2, t^2, t^3, -2*t, 3*t^2, -1]) :

> 'A(t)' = A(t) ; 'B(t)' = B(t) ;

(a) The indefinite integral of the sum of the two matrices is the following ( × ) matrix:

> `indef_int(A+B)` := map(sort, map(int, evalm(A(t)+B(t)), t)) :

> Int(['A(t)' + 'B(t)'], t) = matrix(`indef_int(A+B)`) ;

(b) The sum of the antiderivatives of both matrices is the following ( × ) matrix:

> `indef_int(A) + indef_int(B)` := map(sort, evalm(map(int, A(t), t) + map(int, B(t), t))) :

> Int('A(t)', t) + Int('B(t)', t) = matrix(`indef_int(A) + indef_int(B)`) ;

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

> equal(`indef_int(A+B)`, `indef_int(A) + indef_int(B)`) ;

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

* * *

N.B. The indefinite integral of the product of a scalar (number) and a matrix is equal to the product of the scalar and the matrix antiderivative

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

> A(t) := matrix(2, 3, [t, t^3, -1, t^2, 2*t, -3*t^2]) : 'A(t)' = A(t) ;

and the scalar .

> k := 3 :

(a) The indefinite integral of the product of the scalar and the matrix is the following ( × ) matrix:

> `indef_int(kA)` := map(int, evalm(k * A(t)), t) : Int('k*A(t)', t) = matrix(`indef_int(kA)`) ;

(b) The product of the scalar and the matrix antiderivative is the following ( × ) matrix:

> `k indef_int(A)` := evalm(k * map(int, A(t), t)) : 'k' * Int('A(t)', t) = matrix(`k indef_int(A)`) ;

Both matrices of (a) and (b) are equal by inspection.

* * *

N.B. The indefinite integral of a zero matrix [ 0 ] is the same zero matrix

= [ 0 ]

For example, consider the ( × ) matrix [ 0 ]

> `0` := matrix(2, 3, [0, 0, 0, 0, 0, 0]) : `0` = matrix(`0`) ;

The indefinite integral of the matrix [ 0 ] is the same zero matrix

> `indef_int(0)` := map(int, `0`, t) : Int(`0`, t) = matrix(`indef_int(0)`) ;

* * *

B. Definite integrals

When those elements of a matrix [ A ( ) ] that are functions are all differentiable with respect to in a common interval , then a definite integral of [ A ( ) ] over this interval may be defined.

Consider a ( × ) matrix [ A ( ) ] given as

> A(t) := matrix(2, 3, [a[11](t), a[12](t), a[13](t), a[21](t), a[22](t), a[23](t)]) : 'A(t)' = A(t) ;

The definite integral of [ A ( ) ] over an interval is defined to be the matrix of the same order with elements , viz.

> `def_int(A)` := map(Int, A(t), t=a..b) : Int('A(t)', t=a..b) = matrix(`def_int(A)`) ;

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

> Int(A(t), t=a..b) = matrix(`def_int(A)`) ;

(1) Proper integrals

If neither end point of the integration interval is infinity and the integrand has no singularity in this interval, the integral is proper .

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

> A(t) := matrix(2, 2, [9*exp(-2*t), 3/sqrt(t), sin(t/2), t*cos(t)]) : 'A(t)' = A(t) ;

and compute its integral over the interval , i.e.

> `prp_int(A)` := map(Int, A(t), t=1..3) : Int('A(t)', t=1..3) = matrix(`prp_int(A)`) ;

Exact evaluation of the resultant matrix may be performed using the function map together with the arrow-type procedure including the function value , viz.

> Int('A(t)', t=1..3) = map(x->value(x), `prp_int(A)`) ;

Floating-point evaluation of the exact resultant matrix may be done using any of the following alternative methods.

Method 1 . Using the function evalf and any of the functions evalm , matrix , or op :

> Int('A(t)', t=1..3) = evalf(evalm(`prp_int(A)`)) ; Int('A(t)', t=1..3) = evalf(matrix(`prp_int(A)`)) ; Int('A(t)', t=1..3) = evalf(op(`prp_int(A)`)) ;

Method 2 . Using the function convert together with the form (type) float and any of the functions evalm , matrix , or op :

> Int('A(t)', t=1..3) = convert(evalm(`prp_int(A)`), float) ; Int('A(t)', t=1..3) = convert(matrix(`prp_int(A)`), float) ; Int('A(t)', t=1..3) = convert(op(`prp_int(A)`), float) ;

Method 3 . Using the function map together with the arrow-type procedure including the function evalf :

> Int('A(t)', t=1..3) = map(x->evalf(x), `prp_int(A)`) ;

* * *

N.B. Where the end points and of the integration interval are not given in numerical form, Maple can evaluate the definite integral of a matrix symbolically.

For example, consider the same matrix [ A ( ) ] as before.

The integral of the matrix over the interval is

> `prp_int(A)` := map(Int, A(t), t=a..b) : Int('A(t)', t=a..b) = matrix(`prp_int(A)`) ;

Symbolic evaluation of the matrix integral over the interval yields the following matrix:

> `prp_int(A)` := map(x->value(x), `prp_int(A)`) : Int('A(t)', t=a..b) = matrix(`prp_int(A)`) ;

If numerical values of the integral limits are specified and input in a later phase of computations, e.g.

> a := Pi/4 : b := 3*Pi/2 : 'a' = a ; 'b' = b ;

then exact evaluation of the matrix may be performed as follows:

> `prp_int(A)` := eval(subs('a'=a, 'b'=b, matrix(`prp_int(A)`))) :

> Int('A(t)', t=a..b) = matrix(`prp_int(A)`) ;

Floating-point evaluation of the above matrix yields

> `prp_int(A)` := evalf(matrix(`prp_int(A)`)) : Int('A(t)', t=a..b) = matrix(`prp_int(A)`) ;

* * *

(2) Improper integrals

When the integrand tends to infinity at some point in the interval of integration or the interval itself is infinite in length, the integral is improper .

Maple is also able to compute integrals of matrices if improper integrals are involved. Consider the two cases of improper integrals.

CASE 1 . Improper integral due to infinite interval of integration [ a , ) .

Example 1 . Compute the integral of a ( × ) matrix [ A ( ) ] given as

> A(t) := matrix(2, 3) :

> A(t)[1,1] := exp(-5*t^2) : A(t)[1,2] := t^2*exp(-3*t^2) : A(t)[1,3] := exp(-2*t)*sin(t)/t : A(t)[2,1] := t/(exp(t)+1) : A(t)[2,2] := exp(-2*t^2)*cos(3*t) : A(t)[2,3] := t/(exp(t)-1) :

> A(t) := A(t) : 'A(t)' = A(t) ;

The integral of the matrix over the interval [ , ) is

> `imp_int(A)` := map(Int, A(t), t=0..infinity) : Int('A(t)', t=0..infinity) = matrix(`imp_int(A)`) ;

Symbolic evaluation of the matrix integral yields the following exact matrix:

> `imp_int(A)` := map(x->value(x), `imp_int(A)`) : Int('A(t)', t=0..infinity) = matrix(`imp_int(A)`) ;

Floating-point evaluation of the exact resultant matrix gives

> `imp_int(A)` := evalf(matrix(`imp_int(A)`)) : Int('A(t)', t=0..infinity) = matrix(`imp_int(A)`) ;

Example 2 . Compute the integral of a ( × ) matrix [ B ( ) ] given as

> B(t) := matrix(2, 3) :

> B(t)[1,1] := (sin(t)/t)^2 : B(t)[1,2] := sin(t)*cos(t/2)/t : B(t)[1,3] := 1/(1+t^2) : B(t)[2,1] := tan(t)/t : B(t)[2,2] := cos(t/4)*sin(t/2)/t : B(t)[2,3] := sin(2*t)/t :

> B(t) := B(t) : 'B(t)' = B(t) ;

The integral of the matrix over the interval [ , ) is

> `imp_int(B)` := map(Int, B(t), t=0..infinity) : Int('B(t)', t=0..infinity) = matrix(`imp_int(B)`) ;

Symbolic evaluation of the matrix integral yields the following exact matrix:

> `imp_int(B)` := map(x->value(x), `imp_int(B)`) : Int('B(t)', t=0..infinity) = matrix(`imp_int(B)`) ;

Floating-point evaluation of the exact resultant matrix gives

> `imp_int(B)` := evalf(matrix(`imp_int(B)`)) : Int('B(t)', t=0..infinity) = matrix(`imp_int(B)`) ;

CASE 2 . Improper integral due to discontinuity of the integrand in the interval of integration .

Example 1 . Compute the integral over the interval for a ( × ) matrix [ A ( ) ] given as

> A(t) := matrix(2, 2) :

> A(t)[1,1] := ln(t)/(t-1) : A(t)[1,2] := ln(1+t)/t : A(t)[2,1] := ln(t)/(t^2-1) : A(t)[2,2] := (1+t)*ln(t)/(t-1) :

> A(t) := A(t) : 'A(t)' = A(t) ;

The integral of the matrix over the interval is

> `imp_int(A)` := map(Int, A(t), t=0..1) : Int('A(t)', t=0..1) = matrix(`imp_int(A)`) ;

Notice that each integrand has a singularity at either or in the sense that the integrand tends to infinity as or , respectively.

Symbolic evaluation of the matrix integral yields the following exact matrix:

> `imp_int(A)` := map(x->value(x), `imp_int(A)`) : Int('A(t)', t=0..1) = matrix(`imp_int(A)`) ;

Floating-point evaluation of the exact resultant matrix gives

> `imp_int(A)` := evalf(matrix(`imp_int(A)`)) : Int('A(t)', t=0..1) = matrix(`imp_int(A)`) ;

Example 2 . Compute the integral over the interval for a ( × ) matrix [ B ( ) ] given as

> B(t) := matrix(2, 2) :

> B(t)[1,1] := (sin(4*t)/sin(t))^2 : B(t)[1,2] := ln(cos(t)^4) : B(t)[2,1] := 1/sqrt(t) : B(t)[2,2] := ln(1/cos(t)) :

> B(t) := B(t) : 'B(t)' = B(t) ;

The integral of the matrix over the interval is

> `imp_int(B)` := map(Int, B(t), t=0..Pi/2) : Int('B(t)', t=0..Pi/2) = matrix(`imp_int(B)`) ;

Notice that each integrand has a singularity in the interval at either or in the sense that the integrand tends to infinity as or , respectively.

Symbolic evaluation of the matrix integral yields the following exact matrix:

> `imp_int(B)` := map(x->value(x), `imp_int(B)`) : Int('B(t)', t=0..Pi/2) = matrix(`imp_int(B)`) ;

Floating-point evaluation of the exact resultant matrix gives

> `imp_int(B)` := evalf(matrix(`imp_int(B)`)) : Int('B(t)', t=0..Pi/2) = matrix(`imp_int(B)`) ;

Disregarding the meaningless imaginary parts simplifies the above to

> `imp_int(B)` := evalm(Re(`imp_int(B)`)) : Int('B(t)', t=0..Pi/2) = matrix(`imp_int(B)`) ;

* * *

Proceed to Unit (30) for " Application of the function exp( [ A ] t ) in solving matrix equations ".

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