How to prove if two matrices are similar without explicitly finding the basis matrix? My understanding is that two matrices A and B are similar if and only if there exists some other matrix M such that A = MBM-1.
Is there a way to prove that two matrices are similar without knowing exactly what the basis matrix is? Because obviously if we did, then the "proof" would be a trivial one.
Specifically I'd like to know how the guy in this YouTube video  comes up with similar matrices off the top of his head without explicitly finding the basis matrix.
 A: al eigenvalues are roots of the characteristic polynomial, let's call it $f(x).$
It is possible to test for multiple roots by simply finding the polynomial gcd of $f(x)$ and the derivative $f'(x)$
If $f(x)$ has no multiple roots, and $A,B$ are two matrices that have the same characteristic $f(x),$  then they are similar. In particular, both diagonalize, and the diagonal entries agree.
Now, suppose that there are repeated roots, same for the two matrices. If the minimal polynomials do not agree, the matrices are not similar. The degree of the $(x- \lambda )$ term in the minimal polynomial is the size of the largest Jordan block for that eigenvalue $\lambda$
As the size of the matrices gets larger, there are more and more possibilities of distinct Jordan forms. Which is to say, if the characteristic polynomials and minimal polynomials agree, there may still be room for the matrices to fail to be similar.
A: For two-by-two matrices $A,B$, if the traces and the same, and if the determinants are the same, then at least the characteristic polynomials are the same. Even then, there is still the risk that one has a non-trivial Jordan block, while the other is diagonalizable, and it's maybe not generally easy to distinguish.
An approach for bigger matrices, which already needs some hand computation, is to verify that the trace of $A^n$ is equal to the trace of $B^n$, for $n=1,2,3,\ldots$ up to the size of the matrices. This proves that the characteristic equations are the same. There is still the possibility that the Jordan blocks are different...
