Is every nonsingular matrix diagonally equivalent to a matrix with all distinct eigenvalues?

Abstract

It is shown that a 2 × 2 complex matrix A is diagonally equivalent to a matrix with two distinct eigenvalues iff A is not strictly triangular. It is established in this paper that every 3 × 3 nonsingular matrix is diagonally equivalent to a matrix with 3 distinct eigenvalues. More precisely, a 3 × 3 matrix A is not diagonally equivalent to any matrix with 3 distinct eigenvalues iff det A = 0 and each principal minor of A of order 2 is zero. It is conjectured that for all n ⩾ 2 , an n × n complex matrix is not diagonally equivalent to any matrix with n distinct eigenvalues iff det A = 0 and every principal minor of A of order n - 1 is zero.