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Math / Linear Algebra: A Geometric Approach 2 / Chapter 7.1 / Problem 14

Linear Algebra: A Geometric Approach | 2nd Edition | ISBN: 9781429215213 | Authors: Ted Shifrin, Malcolm Adams

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Let T : Cn Cn be a linear transformation. We say v Cn is a generalized eigenvector of T

Chapter 7, Problem 14

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QUESTION:

Let T : Cn Cn be a linear transformation. We say v Cn is a generalized eigenvector of T with corresponding eigenvalue if v _= 0 and (T I)k(v) = 0 for some positive integer k. Define the generalized -eigenspace E () = {v Cn : v N _ (T I)k _ for some positive integer k}. a. Prove that E() is a subspace of Cn. b. Prove that T ( E()) E().

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QUESTION:

Let T : Cn Cn be a linear transformation. We say v Cn is a generalized eigenvector of T with corresponding eigenvalue if v _= 0 and (T I)k(v) = 0 for some positive integer k. Define the generalized -eigenspace E () = {v Cn : v N _ (T I)k _ for some positive integer k}. a. Prove that E() is a subspace of Cn. b. Prove that T ( E()) E().

ANSWER:

Step 1 of 3

It is given that is a linear transformation.

And, is a generalized eigenvector of with corresponding eigenvalue if and for some positive integer .

The generalized -eigenspace is defined as,

.

To prove that,

a. is a subspace of .

b.

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