Congruence is an equivalence relation. U H U for some non-singular upper triangular U . It is also known as the direct product or tensor product of A and B . Similar matrices represent the same linear transformation in a different basis. Similarity implies equivalence .
Similar matrices have the same trace, determinant, eigenvalues and characteristic equation.
Unitary similarity implies both similarity and conjunctivity . If A and B are real, they are orthogonally similar . Orthogonal similarity implies both similarity and congruence . Equivalence is an equivalence relation. A and B, are equivalent iff they have the same rank.