  The singular values $\sigma >1$ of an $n \times n$ involutory matrix $A$
appear in pairs $(\sigma, \frac{1}{\sigma}),$ while the singular values $\sigma
= 1$ may appear in pairs $(1,1)$ or by themselves. The left and right singular
vectors of pairs of singular values are closely connected. This link is used to
reformulate the singular value decomposition (SVD) of an involutory matrix as
an eigendecomposition. This displays an interesting relation between the
singular values of an involutory matrix and its eigenvalues. Similar
observations hold for the SVD, the singular values and the coneigenvalues of
(skew-)coninvolutory matrices.

Numerical Analysis;Numerical Analysis