.
MATRICES AND MATRIX OPERATIONS: Unit 28
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
-------------------------------------------------------------------
(28) Limits of matrices comprising functions
OBJECTIVES :
• To provide a definition of the limit of a matrix in the form of a symbolic expression.
• To point out that Maple can compute limits of matrices where indeterminate forms are involved.
• To exemplify computations of limits of matrices for all the types of the indeterminate forms.
• To provide five examples of limits of matrices containing functions as their variable tends to infinity .
•
To provide five examples of limits of matrices containing functions as their variable tends to
.
> restart :
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 the limit of such a matrix.
The limit of a matrix [
A
(
)
] whose elements are functions of a
real
variable
, as
approaches
, is defined to be the matrix of the same order with elements
,t = a)](images7/Matrices-Unit287.gif){kind=small}
.
For example, 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) ;
, a[12](t), a[13](t)], [a[2...](images7/Matrices-Unit2811.gif)
The
limit
function applied to each element
](images7/Matrices-Unit2812.gif){kind=small}
of the matrix [
A
(
)
] attempts to compute the limiting value of
](images7/Matrices-Unit2814.gif){kind=small}
as
approaches a given value of the variable, viz.
> `limit(A)` := map(Limit, A(t), t=a) : Limit('A(t)', t=a) = matrix(`limit(A)`) ;
,t = a),...](images7/Matrices-Unit2816.gif)
This operation can be displayed in "like-in-a-book" form, viz.
> Limit(A(t), t=a) = matrix(`limit(A)`) ;
, a[12](t), a[13](t)], [a[21...](images7/Matrices-Unit2817.gif)
In the limit function, an optional parameter may be specified in one of the forms: left , right , real , complex , e.g.
> Limit(f(t), t=a, left) ;
If the parameter is left or right , the limit is a directional limit, taken from the left or right of the limit point a , respectively.
If no additional parameter is specified,
Maple
considers the limit
real
and
bidirectional
. Where the limit point
a
is
infinity
or
-infinity
, the limit of the function is taken from the left to
or from the right to
,
respectively.
Apart from the matrices whose limits are obtainable by computing
for each element
,
Maple
can compute limits where substitution
results in indeterminate forms of type
,
,
,
,
,
,
and
.
In computing such limits,
Maple
uses
L’H
ô
pital’
s
rule. The following examples include all types of the indeterminate forms.
Examples of limits of matrices containing functions as their variable tends to infinity
Example 1
Compute
for a
(
×
)
matrix [
A
(
)
] given as
> a[11](t) := (1+1/t)^t : a[12](t) := (1+2/t)^t : a[21](t) := (t/(1+t))^t : a[22](t) := sin((t/(t-1))^t) :
> A(t) := matrix([ [a[11](t), a[12](t)], [a[21](t), a[22](t)]]) : 'A(t)' = A(t) ;
![A(t) = matrix([[(1+1/t)^t, (1+2/t)^t], [(t/(1+t))^t...](images7/Matrices-Unit2835.gif)
> `lim A(t)` := map(limit, A(t), t=infinity) : Limit('A(t)', t=infinity) = matrix(`lim A(t)`) ;
![Limit(A(t),t = infinity) = matrix([[exp(1), exp(2)]...](images7/Matrices-Unit2836.gif)
Floating-point evaluation yields
> `lim A(t)` := evalf(matrix(`lim A(t)`)) : Limit('A(t)', t=infinity) = matrix(`lim A(t)`) ;
Example 2
Compute
for a
(
×
)
matrix [
B
(
)
] given as
> b[11](t) := (t+(-1)^t)/(t-(-1)^t) : b[12](t) := (3^(t+2)+5^(t+2))/(3^t-5^t) : b[13](t) := (2*t-1)/(3*t+1) : b[21](t) := t*(sqrt(t^2+3) - t) : b[22](t) := t*sin(1/t) : b[23](t) := (1-t)/(1+t) :
> B(t) := matrix([ [b[11](t), b[12](t), b[13](t)], [b[21](t), b[22](t), b[23](t)]]) : 'B(t)' = B(t) ;
> `lim B(t)` := map(limit, B(t), t=infinity) : Limit('B(t)', t=infinity) = matrix(`lim B(t)`) ;
![Limit(B(t),t = infinity) = matrix([[1, -25, 2/3], [...](images7/Matrices-Unit2843.gif)
Example 3
Compute
for a
(
×
)
matrix [
E
(
)
] given as
> e[11](t) := sqrt(1/t+10) - sqrt(1/t-81) : e[12](t) := (1-t^3)/t^2 : e[21](t) := (1/t +1)^2/sin(1/(t-1)) : e[22](t) := sqrt(sin(1/t)+4) - sqrt(sin(1/t)-3) :
> E(t) := matrix([ [e[11](t), e[12](t)], [e[21](t), e[22](t)]]) : 'E(t)' = E(t) ;
> `lim E(t)` := map(limit, E(t), t=infinity) : Limit('E(t)', t=infinity) = matrix(`lim E(t)`) ;
Notice the complex-valued limits of the diagonal elements in the above matrix .
Example 4
Compute
for a
(
×
)
matrix [
F
(
)
] given as
> F(t) := matrix(2, 2) :
> F(t)[1,1] := (t-1)*ln(t^2)/t^2 : F(t)[1,2] := ln(3+2^t)/(4*t+1) : F(t)[2,1] := ln(3*t+1)-ln(2*t+5) : F(t)[2,2] := ln(1+2*exp(t))/t :
> F(t) := F(t) : 'F(t)' = F(t) ;
> `lim F(t)` := map(limit, F(t), t=infinity) : Limit('F(t)', t=infinity) = matrix(`lim F(t)`) ;
![Limit(F(t),t = infinity) = matrix([[0, 1/4*ln(2)], ...](images7/Matrices-Unit2855.gif)
or
> `lim F(t)` := map(combine, `lim F(t)`) : Limit('F(t)', t=infinity) = matrix(`lim F(t)`) ;
![Limit(F(t),t = infinity) = matrix([[0, 1/4*ln(2)], ...](images7/Matrices-Unit2856.gif)
Floating-point evaluation yields
> `lim F(t)` := evalf(matrix(`lim F(t)`)) : Limit('F(t)', t=infinity) = matrix(`lim F(t)`) ;
Example 5
Compute
for a
(
×
)
matrix [
G
(
)
] given as
> G(t) := matrix(2, 2) :
> G(t)[1,1] := (2*t^2+4*t-3)/(5*t^2-6*t+1) : G(t)[1,2] := (t^3-2)/(2*t^3+3*t-4) : G(t)[2,1] := (5*t+3)/(2*t-7) : G(t)[2,2] := (3*t^2+5*t-4)/(5*t^2-t+7) :
> G(t) := G(t) : 'G(t)' = G(t) ;
> `lim G(t)` := map(limit, G(t), t=infinity) : Limit('G(t)', t=infinity) = matrix(`lim G(t)`) ;
![Limit(G(t),t = infinity) = matrix([[2/5, 1/2], [5/2...](images7/Matrices-Unit2863.gif)
* * *
Examples of limits of matrices containing functions as their variable approaches zero
Example 1
Compute
for a
(
×
)
matrix [
A
(
)
] given as
> a[11](t) := 2*(sin(t/4))^3/t^3 : a[12](t) := (t-sin(t))^2/t^3 : a[13](t) := (1-cos(2*t))/t^2 : a[21](t) := ((t^2-t+4)/(t^2-t+1))^(sin(2*t)/t) : a[22](t) := (sqrt(1+t)-1)/t : a[23](t) := ((1-cos(3*t))/t^2)^(sin(2*t)/t) :
> A(t) := matrix([ [a[11](t), a[12](t), a[13](t)], [a[21](t), a[22](t), a[23](t)] ]) : 'A(t)' = A(t) ;
> `lim A(t)` := map(limit, A(t), t=0) : Limit('A(t)', t=0) = matrix(`lim A(t)`) ;
![Limit(A(t),t = 0) = matrix([[1/32, 0, 2], [16, 1/2,...](images7/Matrices-Unit2869.gif)
Example 2
Compute
for a
(
×
)
matrix [
B
(
)
] given as
> b[11](t) := ln(sin(2*t))/ln(sin(t)) : b[12](t) := sin(2*t)/t : b[13](t) := t^sin(t) : b[21](t) := t^t : b[22](t) := t/sin(3*t) : b[23](t) := (sin(t))^t :
> B(t) := matrix([ [b[11](t), b[12](t), b[13](t)], [b[21](t), b[22](t), b[23](t)]]) : 'B(t)' = B(t) ;
> `lim B(t)` := map(limit, B(t), t=0) : Limit('B(t)', t=0) = matrix(`lim B(t)`) ;
![Limit(B(t),t = 0) = matrix([[1, 2, 1], [1, 1/3, 1]]...](images7/Matrices-Unit2875.gif)
Example 3
Compute
for a
(
×
)
matrix [
E
(
)
] given as
> E(t) := matrix(2, 3) :
> E(t)[1,1] := (tan(t)-t)/t^3 : E(t)[1,2] := (1-2*sin(t)^2-cos(t)^3)/(5*t^2) : E(t)[1,3] := sinh(t)/t : E(t)[2,1] := sin(t)^2/t^2 : E(t)[2,2] := (sinh(t)-t)/t^3 : E(t)[2,3] := (tan(t)*arctan(t)-t^2)/t^6 :
> E(t) := E(t) : 'E(t)' = E(t) ;
> `lim E(t)` := map(limit, E(t), t=0) : Limit('E(t)', t=0) = matrix(`lim E(t)`) ;
![Limit(E(t),t = 0) = matrix([[1/3, -1/10, 1], [1, 1/...](images7/Matrices-Unit2881.gif)
Example 4
Compute
for a
(
×
)
matrix [
F
(
)
] given as
> F(t) := matrix(2, 3) :
> F(t)[1,1] := (t-sin(t))/(t-tan(t)) : F(t)[1,2] := (sin(t)-t*cos(t))/t^3 : F(t)[1,3] :=t^3/(sin(t)-t) : F(t)[2,1] := (tan(t)-sin(t))/t^3 : F(t)[2,2] := (tan(t)-t)/(t-sin(t)) : F(t)[2,3] := (exp(t)+exp(-t)-2)/(2*cos(2*t)-2) :
> F(t) := F(t) : 'F(t)' = F(t) ;
> `lim F(t)` := map(limit, F(t), t=0) : Limit('F(t)', t=0) = matrix(`lim F(t)`) ;
![Limit(F(t),t = 0) = matrix([[-1/2, 1/3, -6], [1/2, ...](images7/Matrices-Unit2887.gif)
Example 5
Compute
for a
(
×
)
matrix [
G
(
)
] given as
> G(t) := matrix(2, 3) :
> G(t)[1,1] := (sinh(t)-tanh(t))/t^3 : G(t)[1,2] := (t+sin(t))/(t^2+t) : G(t)[1,3] := (ln(3*sin(t))-ln((1+t)*sin(t)))/(2*exp(t)-1) : G(t)[2,1] := (cot(t)-1/t)/(coth(t)-1/t) : G(t)[2,2] := (exp(sin(t))-t-1)/t^2 : G(t)[2,3] := (4^t-2^t-t*(ln(4)-ln(2)))/t^2 :
> G(t) := G(t) : 'G(t)' = G(t) ;
> `lim G(t)` := map(limit, G(t), t=0) : Limit('G(t)', t=0) = matrix(`lim G(t)`) ;
![Limit(G(t),t = 0) = matrix([[1/2, 2, ln(3)], [-1, 1...](images7/Matrices-Unit2893.gif)
Floating-point evaluation yields
> `lim G(t)` := evalf(matrix(`lim G(t)`)) : Limit('G(t)', t=0) = matrix(`lim G(t)`) ;
* * *
Proceed to Unit (29) for " Integration of matrices comprising functions ".
-------------------------------------------------------------------