Answered: 5) Suppose TE L(V) is invertible. (a)…

5) Suppose TE L(V) is invertible. (a) Suppose 2 E F with 2 # 0. Prove that 2 is an eigenvalue of T if and only if is an eigenvalue of T-1 (b) Prove that T and T-1 have the same eigenvectors.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.4: The Singular Value Decomposition

Problem 26EQ

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Question

100%

5) Suppose T E L(V) is invertible.
(a) Suppose 2 E F with 2 + 0. Prove that 2 is an eigenvalue of T if and only if is an
eigenvalue of T-1.
(b) Prove that T and T-1 have the same eigenvectors.

Expert Solution

Step 1

Note: Every linear transformation can be represented as a matrix. Let T be a linear transformation from Rⁿ to R^m. Then the matrix associated with T is of order m*n.

Also the rank(T) is equal to the rank of the matrix associated with T.