[SOLVED] 18.06- Exercises on similar matrices and Jordan form

Description

Problem 28.1: (6.6 #12. Introduction to Linear Algebra: Strang) These Jordan matrices have eigenvalues 0, 0, 0, 0. They have two eigenvectors; one from each block. However, their block sizes don’t match and they are not similar:

⎡⎤                                          ⎡⎤

J = ⎢_⎢⎥⎥  and K = ⎢_⎢⎥⎥.

⎣⎦                                          ⎣⎦

For a generic matrix M, show that if JM = MK then M is not invertible and so J is not similar to K.

Problem 28.2:(6.6#20.)Whyarethesestatementsalltrue?

2. If A is similar to B then A² is similar to B².
3. A² and B² can be similar when A and B are not similar (try _λ =0,)
4.                          c)  is similar to .
5.                          d)  is not similar to .
6. e) Given a matrix A, let B be the matrix obtained by exchanging rows 1 and 2 of A and then exchanging columns 1 and 2 of A. Show that A is similar to B.

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