Two n-by-n matrices A and B are called similar if
B = S−1AS
for some invertible n-by-n matrix S.
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