ON THE SIMILARITY OF EQUITABLE MATRICES

Abstract

A π-quasi-upper triangular (resp. π-quasi-lower triangular ) π-equitable matrix is a π-equitable matrix whose quotient matrix is upper triangular (resp. lower triangular). A π-quasi-diagonal π-equitable matrix is a π-equitable matrix whose quotient matrix is diagonal. Given matrices A, B ∈ Mn, we say that A and B are similar in a subset S ⊆ Mn if B = P −1AP for some invertible P ∈ S .
Let π be a partition of n, and let A ∈ Mπ . We develop an efficient method for constructing a π-quasi-upper triangular matrix similar to A within Mπ . When the quotient matrix of A is diagonalizable, we further obtain a π-quasi-diagonal matrix similar to A in the same ring.
The method is efficient because the similarity problem for a π-equitable matrix reduces to the much smaller quotient matrix, which can be triangularized or diagonalized directly. The resulting structure is then lifted to a corresponding similarity transformation in Mπ , yielding a quasi-triangular or quasi-diagonal form without ever leaving the class of π- equitable matrices.