22. a. A is diagonalizable if A has n eigenvectors. b. If A is diagonalizable, then A has n distinct eigenvalues. If AP = PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A. c. d. If A is invertible, then A is diagonalizable.
22. a. A is diagonalizable if A has n eigenvectors. b. If A is diagonalizable, then A has n distinct eigenvalues. If AP = PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A. c. d. If A is invertible, then A is diagonalizable.
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
22 pl
![d
at
15.
0
17.
19.
4
-2 <-2
1
0
5
0
7 4
2
0
040
-3
3
0
52
0
5
0
1
2
0
16
8
85
-5
9
-2
0
2
16.
18.
a. =
20.
la
110
-7
6
12
0
-4
402
S
046O
-6
-3
635
4
-16
13 -2
16
1
D N O O
0
0
2
0
0 0
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
and some
matrix
11.1. A is diagonalizable if A- PDP-1 for some matrix D
002
trix with three e
two-dimensional, and one of L
dimensional. Is it possible L
Justify your answer.
is A-1
27. Show that if A is both diagor
28. Show that if A has n linearly
so does AT. [Hint: Use the
29. A factorization A = PDP-
for the matrix A in Exam
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,
counting multiplicities.
d. If A is diagonalizable, then A is invertible.
22. a. A is diagonalizable if A has n eigenvectors.
b. If A is diagonalable, then A has n distinct eigenvalues.
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.
the information in Examp
A = P₁D₁P₁¹.
30. With A and D as in Exan
to the P in Example 2, s
31. Construct a nonzero 2:
diagonalizable.
32. Construct a nondiagon
but not invertible.
[M] Diagonalize the matri
trix program's eigenvalue
then compute bases for th
33.
TTTTTT-
23. A is a 5 x 5 matrix with two eigenvalues. One eigenspace 36.
is three-dimensional, and the other eigenspace is two-
dimensional. Is A diagonalizable? Why?
4
0
-1 -2
4
18
0
1
-6 4
5-2
35. -8 12 -3
6-2
8
8 -18
15
1
41228
0
4
0
6
9 20](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22b1319c-242c-4915-8f56-7c035e43f773%2Fb62dde9f-7a78-44d5-855a-9b16171f55bd%2Fbbwsocb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:d
at
15.
0
17.
19.
4
-2 <-2
1
0
5
0
7 4
2
0
040
-3
3
0
52
0
5
0
1
2
0
16
8
85
-5
9
-2
0
2
16.
18.
a. =
20.
la
110
-7
6
12
0
-4
402
S
046O
-6
-3
635
4
-16
13 -2
16
1
D N O O
0
0
2
0
0 0
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
and some
matrix
11.1. A is diagonalizable if A- PDP-1 for some matrix D
002
trix with three e
two-dimensional, and one of L
dimensional. Is it possible L
Justify your answer.
is A-1
27. Show that if A is both diagor
28. Show that if A has n linearly
so does AT. [Hint: Use the
29. A factorization A = PDP-
for the matrix A in Exam
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,
counting multiplicities.
d. If A is diagonalizable, then A is invertible.
22. a. A is diagonalizable if A has n eigenvectors.
b. If A is diagonalable, then A has n distinct eigenvalues.
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.
the information in Examp
A = P₁D₁P₁¹.
30. With A and D as in Exan
to the P in Example 2, s
31. Construct a nonzero 2:
diagonalizable.
32. Construct a nondiagon
but not invertible.
[M] Diagonalize the matri
trix program's eigenvalue
then compute bases for th
33.
TTTTTT-
23. A is a 5 x 5 matrix with two eigenvalues. One eigenspace 36.
is three-dimensional, and the other eigenspace is two-
dimensional. Is A diagonalizable? Why?
4
0
-1 -2
4
18
0
1
-6 4
5-2
35. -8 12 -3
6-2
8
8 -18
15
1
41228
0
4
0
6
9 20
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