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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?
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  • $\begingroup$ In general if $f(\cdot)$ is analytic then spectrum of $f(A)$ is $f(\lambda_i)$. See Spectral Mapping Theorem. $\endgroup$
    – obareey
    Apr 25 at 7:37

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