Principally normal matrices (Q1938717)

It is said that an \(n \times n\) complex matrix \(A\) is principally normal \((PN)\) if all its principal submatrices are normal, and \(k\)-principally normal \((k-PN)\), \(1 \leq k \leq n\), if all of its principal submatrices of order less than or equal to \(k\) are normal. In this paper, the characterization of principally normal matrices is given. A relationship is established between entries of an irreducible essentially Hermitian \(PN\) matrix. It is also proved the Cauchy interlacing theorem for a class of normal matrices including the Hermitian matrices, i.e., that the eigenvalues of any principal submatrix of an irreducible \(PN\) matrix interlace those of the matrix itself on a line in the complex plane that passes through the origin.