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Let V be a vector space over F. Let T E L(V). Suppose PE L(V) is invertible. (a) Prove that T and P 'TP have the same eigenvalues. (Hint: TPP ' =TI = T.) (b) What is the relationship between the eigenvectors of T and the eigenvectos of P !TP?

Let V be a vector space over F. Let T E L(V). Suppose PE L(V) is invertible. (a) Prove that T and P 'TP have the same eigenvalues. (Hint: TPP ' =TI = T.) (b) What is the relationship between the eigenvectors of T and the eigenvectos of P !TP?

Linear Algebra: A Modern Introduction
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 36EQ: Consider again the matrix A in Exercise 35. Give conditions on a, b, c, and d such that A has two...
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Let VV be a vector space over FF. Let T∈L(V)T∈L(V). Suppose P∈L(V)P∈L(V) is invertible.

(a) Prove that TT and P−1TPP−1TP have the same eigenvalues. (Hint: TPP−1=TI=TTPP−1=TI=T.)

(b) What is the relationship between the eigenvectors of TT and the eigenvectos of P−1TPP−1TP?

Let V be a vector space over F. Let T E L(V). Suppose PE L(V) is invertible. (a) Prove that T and P TP have the same eigenvalues. (Hint: TPP 1 =TI = T.) (b) What is the relationship between the eigenvectors of T and the eigenvectos of P 'TP?
Transcribed Image Text:Let V be a vector space over F. Let T E L(V). Suppose PE L(V) is invertible. (a) Prove that T and P TP have the same eigenvalues. (Hint: TPP 1 =TI = T.) (b) What is the relationship between the eigenvectors of T and the eigenvectos of P 'TP?
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