similar matrices

Square matrices $A,B$ with $B = P^{-1}AP$ for some invertible $P$ .

Square matrices $A$ and $B$ are similar if $B=P^{-1}AP$ for some invertible $P$ — the same matrix, read in a different basis built from $P$ 's columns. Similarity is an equivalence relation (reflexive, symmetric, transitive), and any basis-independent fact — eigenvalues, trace, determinant, rank — must come out identical for similar matrices.