Problem #6: LetProblem #6(a):8A = 0 90(A) 11(a) Find a matrix P that diagonalizes A.(b) Find P-¹AP [using your answer from (a)].1(G) 01 0 8101 1Problem #6(b):0 1000 90 10100 01Select v10 0(A) 0 90 0 99 0 0(F) 0 9 00 0 10Select1 0 8(B) 1 10080 (B) 001(H) 10(G) 00 -110 (C) 10 81 00 101008 090 00010-1009010101 -19 0(C) 0 9008(H) 0'808009 00 0 910(D) 1 1109106(E) 00100 -8(D) 0 10 00 081 011 1(E)910 (F) 001000010080008100 1

(.) Given matrix is, (.) Diagonalization : A square matrix is said to be diagonalizable…

Answered: Problem #6: Let Problem #6(a): A = 0…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Problem #6: Let
Problem #6(a):
8
A = 0 9
0
(A) 1
1
(a) Find a matrix P that diagonalizes A.
(b) Find P-¹AP [using your answer from (a)].
1
(G) 0
1 0 8
1
0
1 1
Problem #6(b):
0 10
0
0 9
0 10
1
0
0 0
1
Select v
10 0
(A) 0 9
0 0 9
9 0 0
(F) 0 9 0
0 0 10
Select
1 0 8
(B) 1 1
0
0
8
0 (B) 0
0
1
(H) 1
0
(G) 0
0 -1
1
0 (C) 1
0 8
1 0
0 1
0
10
0
8 0
9
0 0
0
0
10
-10
0
9
0
10
1
0
1 -1
9 0
(C) 0 9
0
0
8
(H) 0
'80
8
0
0
9 0
0 0 9
1
0
(D) 1 1
1
0
9
106
(E) 0
0
10
0 -8
(D) 0 10 0
0 0
8
1 0
1
1 1
(E)
9
1
0 (F) 0
0
10
0
0
0
10
0
800
0
8
1
0
0 1

Expert Solution

Step 1: We give definition of diagonalization of a matrix.

(.) Given matrix is,

$A space equals space open square brackets table row 8 0 10 row 0 9 0 row 0 0 9 end table close square brackets$

(.) Diagonalization : A square matrix $apostrophe A apostrophe$ is said to be diagonalizable if there exists non - singular matrix $apostrophe P apostrophe$ such that $P to the power of negative 1 end exponent A P equals D$ .

Where $D$ is a diagonal matrix whose diagonal entries are the eigen values of matrix $A$ .

And matrix $P$ is formed by taking eigen vectors of matrix $A$ as columns.

(.) A square matrix $apostrophe A apostrophe$ is diagonalizable if and only if

$A. M. space equals space G. M. space space space for all space lambda$

here $lambda$ is an eigen value.

(.) Algebraic multiplicity ( A.M. ) : The number of times an eigen value occurs is called algebraic multiplicity.

(.) Geometric multiplicity ( G.M.) : G.M. of an eigen value $apostrophe lambda apostrophe$ of an $apostrophe n cross times n apostrophe$ matrix $apostrophe A apostrophe$ is given by,

$G. M. space o f space apostrophe lambda apostrophe space equals space n minus r a n k open parentheses A minus lambda I close parentheses$

here $apostrophe I apostrophe$ is an $apostrophe n cross times n apostrophe$ identity matrix.

(.) If A.M. of an eigen value $lambda$ is $apostrophe 1 apostrophe$ then G.M. of $apostrophe lambda apostrophe$ is also $apostrophe 1 apostrophe$ ,

$rightwards double arrow space space space space 1 space less or equal than space A. M. space less or equal than space G. M.$