  The centralizer of a matrix in a full matrix algebra is called a principal
centralizer matrix algebra. Characterizations are presented for principal
centralizer matrix algebras to be Morita equivalent, almost $\nu$-stable
derived equivalent, derived equivalent, and stably equivalent of Morita type,
respectively, in terms of new equivalence relations on square matrices. These
equivalence relations on matrices are introduced in a natural way by their
elementary divisors. Thus the categorical equivalences are reduced to questions
in linear algebra. Consequently, principal centralizer matrix algebras of
permutation matrices are Morita equivalent if and only if they are derived
equivalent. Moreover, two representation-finite, principal centralizer matrix
algebras over a perfect field are stably equivalent of Morita type if and only
if they are stably equivalent. Further, derived equivalences between the
principal centralizer matrix algebras of permutation matrices induce the ones
of their $p$-regular parts and $p$-singular parts of the given permutations.
