Let A = -1 (a) Find the characteristic polynomial and the eigenvalues of A over the complex numbers. (b) Find a basis of the eigenspace of one of the eigenvalues of A over the complex numbers. (c) Without any more computation, determine a basis for the eigenspace of the other eigenvalue of A over the complex numbers. Explain briefly how you determined your answer. (d) Find a matrix P and a matrix D such that P-¹AP = D. [Hint: you do not have to compute P¹. Note that you can check your work by checking AP = PD.]
Let A = -1 (a) Find the characteristic polynomial and the eigenvalues of A over the complex numbers. (b) Find a basis of the eigenspace of one of the eigenvalues of A over the complex numbers. (c) Without any more computation, determine a basis for the eigenspace of the other eigenvalue of A over the complex numbers. Explain briefly how you determined your answer. (d) Find a matrix P and a matrix D such that P-¹AP = D. [Hint: you do not have to compute P¹. Note that you can check your work by checking AP = PD.]
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.1: Introduction To Eigenvalues And Eigenvectors
Problem 38EQ
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![Let
A =
-1
(a) Find the characteristic polynomial and the eigenvalues of A over the complex numbers.
(b) Find a basis of the eigenspace of one of the eigenvalues of A over the complex numbers.
(c) Without any more computation, determine a basis for the eigenspace of the other eigenvalue of
A over the complex numbers. Explain briefly how you determined your answer.
(d) Find a matrix P and a matrix D such that P-¹AP
=
D. [Hint: you do not have to compute
P¹. Note that you can check your work by checking AP = PD.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff880b47a-a06f-4ca0-be7f-0ccc8d2dab4b%2Faec5c3ce-97b1-4b97-a1de-ba9d80c7cd5f%2F47677aub_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let
A =
-1
(a) Find the characteristic polynomial and the eigenvalues of A over the complex numbers.
(b) Find a basis of the eigenspace of one of the eigenvalues of A over the complex numbers.
(c) Without any more computation, determine a basis for the eigenspace of the other eigenvalue of
A over the complex numbers. Explain briefly how you determined your answer.
(d) Find a matrix P and a matrix D such that P-¹AP
=
D. [Hint: you do not have to compute
P¹. Note that you can check your work by checking AP = PD.]
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