Characterization and Properties of (, )-Symmetric, (, )-Skew Symmetric, and (, )-Conjugate Matrices

Let ∈ ℂ and ∈ ℂ be nontrivial involutions; i.e., = ≠ ± and = ≠ ±. We say that ∈ ℂ is (, )-symmetric ((, )-skew symmetric) if = ( = −).We give an explicit representation of an arbitrary ()-symmetric matrix in terms of matrices and associated with and and associated with . If = * then the least squares problem for can be solved by solving the independent least squares problems for and . If, in addition, either rank() = or * = , then can be expressed in terms of and . If = * and = * then a singular value decomposition of can obtained from singular value decompositions of and . Similar results hold for ()-skew symmetric matrices.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. If = the least squares problem for the complex matrix reduces to two least squares problems for the real matrix . If, in addition, either rank() = or = , then can be obtained from . If both = and = , a singular value decomposition of can be obtained from a singular value decomposition of