25. A is a 4 x 4 matrix with three eigenvalues. One eigenspace is one-dimensional, and one of the other eigenspaces is two- dimensional. Is it possible that A is not diagonalizable? Justify your answer.
25. A is a 4 x 4 matrix with three eigenvalues. One eigenspace is one-dimensional, and one of the other eigenspaces is two- dimensional. Is it possible that A is not diagonalizable? Justify your answer.
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
25 please simple and short explanation
![201
ces.
udy
fore
D
24. A is a 3 x 3 matrix with two eigenvalues. Each eigenspace is
one-dimensional. Is A diagonalizable? Why?
25. A is a 4 x 4 matrix with three eigenvalues. One eigenspace
is one-dimensional, and one of the other eigenspaces is two-
dimensional. Is it possible that A is not diagonalizable?
Justify your answer.
26. A is a 7 x 7 matrix with three eigenvalues. One eigenspace is
two-dimensional, and one of the other eigenspaces is three-
dimensional. Is it possible that A is not diagonalizable?
Justify your answer.
175
27. Show that if A is both diagonalizable and invertible, then so
is A-¹.
28. Show that if A has n linearly independent eigenvectors, then
so does AT. [Hint: Use the Diagonalization Theorem.]
0
29. A factorization A = PDP is not unique. Demonstrate this
xample 2. With D₁ =
for the matrix A la
[39].
5
the information in Example 2 to find a matrix P₁ such that
A = P₁D₁P₁²¹.
use
30. With A and D as in Example 2, find an invertible P₂ unequal
to the P in Example 2, such that A = P₂DP₂¹.
31. Construct a nonzero 2 x 2 matrix that is invertible but not
diagonalizable.
32. Construct a nondiagonal 2 x 2 matrix that is diagonalizable
but not invertible.
[M] Diagonalize the matrices in Exercises 33-36. Use your ma-
trix program's eigenvalue command to find the eigenvalues, and
then compute bases for the eigenspaces as in Section 5.1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F22b1319c-242c-4915-8f56-7c035e43f773%2Fa545c1f1-1002-4ff0-bd30-d873a523f8fb%2Fdic9k3m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:201
ces.
udy
fore
D
24. A is a 3 x 3 matrix with two eigenvalues. Each eigenspace is
one-dimensional. Is A diagonalizable? Why?
25. A is a 4 x 4 matrix with three eigenvalues. One eigenspace
is one-dimensional, and one of the other eigenspaces is two-
dimensional. Is it possible that A is not diagonalizable?
Justify your answer.
26. A is a 7 x 7 matrix with three eigenvalues. One eigenspace is
two-dimensional, and one of the other eigenspaces is three-
dimensional. Is it possible that A is not diagonalizable?
Justify your answer.
175
27. Show that if A is both diagonalizable and invertible, then so
is A-¹.
28. Show that if A has n linearly independent eigenvectors, then
so does AT. [Hint: Use the Diagonalization Theorem.]
0
29. A factorization A = PDP is not unique. Demonstrate this
xample 2. With D₁ =
for the matrix A la
[39].
5
the information in Example 2 to find a matrix P₁ such that
A = P₁D₁P₁²¹.
use
30. With A and D as in Example 2, find an invertible P₂ unequal
to the P in Example 2, such that A = P₂DP₂¹.
31. Construct a nonzero 2 x 2 matrix that is invertible but not
diagonalizable.
32. Construct a nondiagonal 2 x 2 matrix that is diagonalizable
but not invertible.
[M] Diagonalize the matrices in Exercises 33-36. Use your ma-
trix program's eigenvalue command to find the eigenvalues, and
then compute bases for the eigenspaces as in Section 5.1.
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