Matrices with multiple symmetry properties: Applications of centrohermitian and perhermitian matrices

Twelve known symmetry patterns of matrices are combined with three modest patterns to form a steiner triple system. We investigate matrices satisfying more than one symmetry pattern. We show how a group of operators on GL(n, ℂ) gives rise to distinct types of matrices which satisfy sets of patterns, and which give unique decompositions of matrices into components of each type. These give a new characterization of normal and unitary matrices. We extend symmetry patterns to vectors to study spectral properties of these matrices. When a (skew) symmetric basis of eigenvectors exist, we can infer symmetry properties of these matrices