On Almost Normal Matrices

An n-by-n matrix is called almost normal if the maximal cardinality of a set of orthogonal eigenvectors is at least n-1. We give several basic properties of almost normal matrices, in addition to studying their numerical ranges and Aluthge transforms. First, a criterion for these matrices to be unitarily irreducible is established, in addition to a criterion for the conjugate transpose of an almost normal matrix to be almost normal and a formula for the rank of the self commutator of an almost normal matrix. We then show that unitarily irreducible almost normal matrices cannot have flat portions on the boundary of their numerical ranges and that the Aluthge transform of an almost normal matrix is never normal when n > 2 and the almost normal matrix is unitarily irreducible and invertible