Determine which of the following matrices are similar: , , , and .
Okay, these matrix computations are getting to be excruciating to do by hand. I’m going to completely wimp out and use a CAS to find the matrices such that are in Jordan canonical form. The good news is that this does not render the proof incomplete, since we can easily verify that is in JCF. I wanted to get my hands dirty (so to speak) actually using the algorithm, but now that I’ve got a handle on it I’ll use a computer.
Note: WolframAlpha can compute the Jordan Canonical Form of a matrix with the syntax ‘jordan form A’ where A is a matrix in the form [[r11,r12,…,r1n],[r21,r22,…,r2n],[rn1,rn2,…,rnn]]. (Example.)
Let , , , and . Evidently then we have , , and . These products are in Jordan canonical form (obviously). So and are similar, and and are not similar to each other or to .