Editing Cayley transform (section)

=== Other matrices ===
One can extend the mapping to [[complex number|complex]] matrices by substituting "[[unitary matrix|unitary]]" for "orthogonal" and "[[skew-Hermitian matrix|skew-Hermitian]]" for "skew-symmetric", the difference being that the transpose (·<sup>T</sup>) is replaced by the [[conjugate transpose]] (·<sup>H</sup>). This is consistent with replacing the standard real [[inner product]] with the standard complex inner product. In fact, one may extend the definition further with choices of [[Hermitian adjoint|adjoint]] other than transpose or conjugate transpose.

Formally, the definition only requires some invertibility, so one can substitute for ''Q'' any matrix ''M'' whose eigenvalues do not include −1. For example,
:<math>
\begin{bmatrix} 0 & -a & ab - c \\ 0 & 0 & -b \\ 0 & 0 & 0 \end{bmatrix}
\leftrightarrow
\begin{bmatrix} 1 & 2a & 2c \\ 0 & 1 & 2b \\ 0 & 0 & 1 \end{bmatrix} .
</math>
Note that ''A'' is skew-symmetric (respectively, skew-Hermitian) if and only if ''Q'' is orthogonal (respectively, unitary) with no eigenvalue −1.