# Eigenvalues properties under Cayley transform

Suppose I have some matrix $$\mathbf{A}\in \mathbb{C}^{n\times n}$$ and Cayley transform defined as $$\begin{equation*} \tilde{\mathbf{A}} = (\mathbf{I} - \mathbf{A})(\mathbf{I} + \mathbf{A})^{-1} \end{equation*}$$ where $$\mathbf{I}$$ is the identity matrix.

1. Can we say something about how spectra $$\sigma(\mathbf{A})$$ and $$\sigma(\tilde{\mathbf{A}})$$ are related in terms of eigenvalues (some closed form formula for instance)?
2. Is it possible to convert from one to other by knowing its minors and determinant?
• In general if $f(\cdot)$ is analytic then spectrum of $f(A)$ is $f(\lambda_i)$. See Spectral Mapping Theorem. Apr 25 at 7:37