Cayley transform for Toeplitz and dual matrices

Abstract

Let an $n\times n$ complex matrix A be such that $I+A$ is invertible. The Cayley transform of A, denoted by $C\left(A\right)$, is defined as$C\left(A\right)={(I+A)}^{-1}(I-A).$ Fallat and Tsatsomeros (2002) [5] and Mondal et al. (2024) [15] studied the Cayley transform of a matrix A in the context of P-matrices, H-matrices, M-matrices, totally positive matrices, positive definite matrices, almost skew-Hermitian matrices, and semipositive matrices. In this paper, the investigation of the Cayley transform is continued for Toeplitz matrices, circulant matrices, unipotent matrices, and dual matrices. An expression of the Cayley transform for dual matrices is established. It is shown that the Cayley transform of a dual symmetric matrix is always a dual symmetric matrix. The Cayley transform of a dual skew-symmetric matrix is discussed. The results are illustrated with examples.