Description
Exercises on similar matrices and Jordan form
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 1 0 0
J = ⎢
⎢
⎣
0 0 0 0
0 0 0 1
⎥
⎥
⎦
and K = ⎢
⎢
⎣
0 0 1 0
0 0 0 0
⎥
⎥
⎦ .
0 0 0 0 0 0 0 0
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.)
3 0 3 1 c) is similar to . 0 4 0 4
3 0 3 1 d) is not similar to . 0 3 0 3
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.
1
18.06SC Linear Algebra
