Franklin's Notes

Cayley Transform

The Cayley Transform of a matrix $A$ is the matrix $(I-A)^{-1}(I+A)$, defined as long as $1$ is not an eigenvalue of $A$. This is an analogue of the fractional linear transformation in the complex plane, which maps the imaginary axis to the unit circle. Similarly, the corresponding matrix transformation transforms skew-Hermitian matrices to unitary matrices .

Proposition 1. The Cayley Transform of a skew-Hermitian matrix is unitary.

Proof. Let $Q=(I-S)^{-1}(I+S)$, where $S$ is skew-Hermitian. First of all, $S$ must have purely imaginary eigenvalues, since $iS$ is Hermitian and must therefore have purely real eigenvalues. This means that $I-S$ is a nonsingular matrix, and $(I-S)^{-1}$ exists, so the Cayley Transform of $S$ is well-defined.

Now, we have that meaning that $Q$ is unitary as claimed! $\blacksquare$

matrices

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