On the Cayley Transform of Matrix Classes

The Cayley transform of a square matrix $A$, defined as $F=(I+A)^{-1}(I-A)$, is tantamount to the factorization $A=(I+F)^{-1}(I-F)$. In this context, Fallat {\it et al.} (Electron. J. Linear Algebra, 9:190-196, 2002) and Mondal {\it et al.} (Linear Algebra Appl., 681:1-20, 2024) studied the Cayley transform of matrix positivity classes, namely, $P$-matrices, positive definite matrices, as well as $H$-matrices, $M$-matrices, and their inverse classes. The Cayley transform of $J$-symplectic, Toeplitz and dual matrices has also been considered in the literature. In this paper, the discussion is extended by examining the Cayley transform of positive semidefinite matrices, $Q^*$-matrices, EP-matrices, weighted-EP matrices, GP matrices, idempotent matrices, $T$-Hermitian matrices, $T$-EP matrices, $S$-skew symmetric matrices, $S$-normal matrices, centrosymmetric matrices, tridiagonal matrices, block triangular matrices, and semiconvergent matrices. The results complement the existing literature and are illustrated with examples. Connections among the matrix classes considered are also discussed and a summary of all results on the matrix Cayley transform known to date is compiled in the form of a table.