A condensed representation of almost normal matrices

In this paper we study the structure of almost normal matrices, that is the matrices for which there exists a rank-one matrix C such that AHA-AAH=CA-AC. Necessary and sufficient conditions for a matrix to belong to the class are given and a canonical representation as a block tridiagonal matrix is shown. The approach is constructive and in the paper it is explained how, starting from a 1×1 or 2×2 matrix we can generate almost normal matrices. Moreover, given an n×n almost normal matrix we can compute the block tridiagonal representation with a finite procedure.