4. Let A be a square matrix. (a) 1² is an eigenvalue of A² with the same eigenvector, v. Show that if A is an eigenvalue of A with corresponding eigenvector v, then (b) What is the characteristic polynomial of A²? Suppose that the characteristic polynomial of A is fa(^) = (A + 2)(A – 3). %3D Suppose that v is an eigenvector of A². Is it necessarily true that v is also (c) an eigenvector of A? If so, then prove this fact; if not, then give an example that shows this is false.
4. Let A be a square matrix. (a) 1² is an eigenvalue of A² with the same eigenvector, v. Show that if A is an eigenvalue of A with corresponding eigenvector v, then (b) What is the characteristic polynomial of A²? Suppose that the characteristic polynomial of A is fa(^) = (A + 2)(A – 3). %3D Suppose that v is an eigenvector of A². Is it necessarily true that v is also (c) an eigenvector of A? If so, then prove this fact; if not, then give an example that shows this is false.
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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Transcribed Image Text:4. Let A be a square matrix.
(a)
1² is an eigenvalue of A? with the same eigenvector, v.
Show that if A is an eigenvalue of A with corresponding eigenvector v, then
(b)
What is the characteristic polynomial of A2?
Suppose that the characteristic polynomial of A is fa(1) = (A + 2)(A – 3).
Suppose that v is an eigenvector of A². Is it necessarily true that v is also
(c)
an eigenvector of A? If so, then prove this fact; if not, then give an example that shows
this is false.
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