Two n-by-n matrices A and B are called similar if $ \! B = P^{-1} A P $ for some invertible n-by-n matrix P.
Similar matrices share many properties:
- Rank
- Determinant
- Trace
- Eigenvalues (though the eigenvectors will in general be different)
- Characteristic polynomial
- Minimal polynomial (among the other similarity invariants in the Smith normal form)
- Elementary divisors
Given two square matrices A and B, how would you tell if they are similar?
- Constructing a $P$ in the definition seems difficult even if we know they are similar, does it? Not to mention, use this way to tell if they are similar.
- Are there some properties of similar matrices that can characterize similar matrices?
Thanks!