# A note on the hyperbolic singular value decomposition without hyperexchange matrices

## TL;DR

This paper introduces a new formulation of the hyperbolic singular value decomposition (HSVD) that simplifies computation by avoiding hyperexchange matrices, using pseudo-unitary matrices, and relates to eigenvalue problems, enhancing its applicability.

## Contribution

The paper presents a novel HSVD formulation that eliminates hyperexchange matrices and redundant parameters, relying solely on pseudo-unitary matrices, and connects HSVD computation to eigenvalue problems.

## Key findings

- Simplifies HSVD computation process
- Reduces the problem to eigenvalue and eigenvector calculations
- Includes ordinary SVD as a special case

## Abstract

We present a new formulation of the hyperbolic singular value decomposition (HSVD) for an arbitrary complex (or real) matrix without hyperexchange matrices and redundant invariant parameters. In our formulation, we use only the concept of pseudo-unitary (or pseudo-orthogonal) matrices. We show that computing the HSVD in the general case is reduced to calculation of eigenvalues, eigenvectors, and generalized eigenvectors of some auxiliary matrices. The new formulation is more natural and useful for some applications. It naturally includes the ordinary singular value decomposition.