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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.

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.

Advanced Engineering Mathematics
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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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.
Transcribed Image Text: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.
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Note: Every linear transformation can be represented as a matrix. Let T be a linear transformation from Rn to Rm. 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.

 

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