21. a. A is diagonalizable if A = PDP-¹ for some matrix D and some invertible matrix P. b. If R" has a basis of eigenvectors of A, then A is diago- nalizable. c. A is diagonalizable if and only if A has n eigenvalues, multiplicities. counting d. If A is diagonalizate, then A is invertible. 1003 27
21. a. A is diagonalizable if A = PDP-¹ for some matrix D and some invertible matrix P. b. If R" has a basis of eigenvectors of A, then A is diago- nalizable. c. A is diagonalizable if and only if A has n eigenvalues, multiplicities. counting d. If A is diagonalizate, then A is invertible. 1003 27
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
21 pl
![d
at
0
he
15.
17
19.
-2 -2
1
(
0
0
0
5-3
3
0
4
0
0
0
0
0
5
0
1
-5
2
0
9
-2
0
2
18.
20.
ī
1
-7
6
12
11
-16
0400
iņn
0-3
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16 1
OONO
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0
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2
In Exercises 21 and 22, A, B, P, and D are nxn matrices.
Mark each statement True or False. Justify each answer. (Study
you try these exercises.)
Theorems 5 and 6 and the examples in this section carefully before
002
21. a. A is diagonalizable if A = PDP-1 for some matrix D
and some invertible matrix P.
b. If R" has a basis of eigenvectors of A, then A is diago-
nalizable.
0
d.
If A is diagonalizate, then A is invertible.
0 2
c. A is diagonalizable if and only if A has n eigenvalues,
counting multiplicities.
22. a. A is diagonalizable if A has n eigenvectors.
it inson
b. If A is diagonalizable, then A has n distinct eigenvalues.
27. Show that if A is b
is A-1
c. If AP = PD, with D diagonal, then the nonzero columns
of P must be eigenvectors of A.
d. If A is invertible, then A is diagonalizable.
23. A is a 5 x 5 matrix with two eigenvalues. One eigenspace
is three-dimensional, and the other eigenspace is two-
dimensional. Is A diagonalizable? Why?
28. Show that if A has
so does AT. [Hint:
29. A factorization A
for the matrix A
the information in
A = P₁D₁P₁
30. With A and D as
to the P in Exam
31. Construct a non:
diagonalizable.
32. Construct a non
but not invertibl
[M] Diagonalize the
trix program's eigem
then compute bases
33.
-6
4
-3
0
-1 -2
4
11
-6
-3
5
35. -8 12
6
36.
81
8-18
4
4
0 1
6 12
9 20
28
15](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22b1319c-242c-4915-8f56-7c035e43f773%2F550d52a3-d585-4a66-9f63-d11e427aa64e%2Fw2fzbl9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:d
at
0
he
15.
17
19.
-2 -2
1
(
0
0
0
5-3
3
0
4
0
0
0
0
0
5
0
1
-5
2
0
9
-2
0
2
18.
20.
ī
1
-7
6
12
11
-16
0400
iņn
0-3
2
4
13 -2
16 1
OONO
wan
0
0
2
In Exercises 21 and 22, A, B, P, and D are nxn matrices.
Mark each statement True or False. Justify each answer. (Study
you try these exercises.)
Theorems 5 and 6 and the examples in this section carefully before
002
21. a. A is diagonalizable if A = PDP-1 for some matrix D
and some invertible matrix P.
b. If R" has a basis of eigenvectors of A, then A is diago-
nalizable.
0
d.
If A is diagonalizate, then A is invertible.
0 2
c. A is diagonalizable if and only if A has n eigenvalues,
counting multiplicities.
22. a. A is diagonalizable if A has n eigenvectors.
it inson
b. If A is diagonalizable, then A has n distinct eigenvalues.
27. Show that if A is b
is A-1
c. If AP = PD, with D diagonal, then the nonzero columns
of P must be eigenvectors of A.
d. If A is invertible, then A is diagonalizable.
23. A is a 5 x 5 matrix with two eigenvalues. One eigenspace
is three-dimensional, and the other eigenspace is two-
dimensional. Is A diagonalizable? Why?
28. Show that if A has
so does AT. [Hint:
29. A factorization A
for the matrix A
the information in
A = P₁D₁P₁
30. With A and D as
to the P in Exam
31. Construct a non:
diagonalizable.
32. Construct a non
but not invertibl
[M] Diagonalize the
trix program's eigem
then compute bases
33.
-6
4
-3
0
-1 -2
4
11
-6
-3
5
35. -8 12
6
36.
81
8-18
4
4
0 1
6 12
9 20
28
15
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