How do I get the similarity invariants from a jordan canonical form?

Question

How do I get the similarity invariants out of a Jordan canonical form? Do I have to transform it into a Fröbenius form and then read them from the top left corner and bottom right corner or can I directly get it from the Jordan Normal Form?

Answer 1

Two matrices of Jordan normal form are similar iff they have the same block structure, up to permutation, i.e. iff there is a one-to-one correspondence between their Jordan blocks.

For example, $$\pmatrix{2&0&0\\1&2&0\\0&1&2\\&&&2&0\\&&&1&2\\&&&&&3}\ \sim\ \pmatrix{3\\&2&0\\&1&2\\&&&2&0&0\\&&&1&2&0\\&&&0&1&2}$$