Characterization and Properties of Matrices with Generalized Symmetry or Skew Symmetry

Let ∈ ℂ be a nontrivial involution; i.e., = ≠ ± We say that ∈ ℂ is -symmetric (-skew symmetric) if = ( = − ).There are positive integers and with + = and matrices ∈ ℂ and ∈ ℂ such that = , = , = , and = −. We give an explicit representation of an arbitrary -symmetric matrix in terms of and , and show that solving = and the eigenvalue problem for reduce to the corresponding problems for matrices Ȉ ℂ and Ȉ ℂ. We also express in terms of and . Under the additional assumption that = , we show that Moore–Penrose inversion and singular value decomposition of reduce to the corresponding problems for and . We give similar results for -skew symmetric matrices. These results are known for the case where however, our proofs are simpler even in this case.We say that ∈ ℂ is -conjugate if where ∈ ℝ and = ≠ . In this case () is -symmetric and () is -skew symmetric, so our results provide explicit representations for -conjugate matrices in terms of and , which are now in ℝ and ℝ respectively. We show that solving = , inverting , and the eigenvalue problem for reduce to the corresponding problems for a related matrix ∈ ℝ. If = this is also true for Moore–Penrose inversion and singular value decomposition of