On the Matrices of Central Linear Mappings

TL;DR

This paper characterizes when a central linear mapping between projectively embedded Euclidean spaces can be decomposed into a projection and a similarity, based on the multiplicity of the least singular value of an associated matrix.

Contribution

It provides a necessary and sufficient condition for the decomposability of central linear mappings in terms of singular value multiplicity.

Findings

Decomposability depends on the multiplicity of the least singular value.

A specific matrix derived from the mapping determines this property.

The condition links geometric decompositions to matrix singular values.

Abstract

We show that a central linear mapping of a projectively embedded Euclidean $n$-space onto a projectively embedded Euclidean $m$-space is decomposable into a central projection followed by a similarity if, and only if, the least singular value of a certain matrix has multiplicity $\ge 2m-n+1$. This matrix is arising, by a simple manipulation, from a matrix describing the given mapping in terms of homogeneous Cartesian coordinates.