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:
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0 1
0 0
0 0
0 0
0 0
0 1
0 0
0 0
⎡⎤ ⎡⎤
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.) Why are these statements all true?
a) If A is similar to B then A2 is similar to B2.
b) A2 and B2 can be similar when A and B are not similar (try λ = 0, 0.)
c) is similar to .
d) is not similar to .
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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