Cosquares:complex and otherwise
Charles S. Ballantine, Ferdinand Georg Frobenius · Linear and Multilinear Algebra · 1978
Matrices of the form S ∗−1 S and matrices A = TA ∗−1 T −1for some T = ∊ T ∗ are characterized in (1) the usual complex case (where M ∗ denotes the conjugate transpose of M), (2) The usual real case (where M ∗ = M ′ is the transpose of M), and in the respective generalizations, (1) an arbitrary field with involution of order two and (2) an arbitrary field. Also in case (1') matrices S ∗−1 S withS = S ′ are characterized, giving a slight improvement on (and independent proof of) Djokovic's recent characterization of matrices in particular, every nonsingular matrix M be factored as M = RS with R= [Rbar] and S = S ′, In case (2'), alternating matrices T are also considered (when char = 2)