# A note on the singular value decomposition of (skew-)involutory and (skew-)coninvolutory matrices

## TL;DR

This paper explores the singular value decomposition of involutory and (skew-)coninvolutory matrices, revealing relationships between their singular values, eigenvalues, and coneigenvalues, and reformulating SVD as eigendecomposition.

## Contribution

It introduces a novel reformulation of SVD for involutory matrices as eigendecomposition, highlighting the connection between singular values and eigenvalues for these matrices.

## Key findings

- Singular values of involutory matrices come in reciprocal pairs.
- SVD can be reformulated as eigendecomposition for involutory matrices.
- Similar properties hold for (skew-)coninvolutory matrices.

## Abstract

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.