The linear targeting problem

TL;DR

This paper investigates conditions under which specific types of matrices A can be found to satisfy the linear equation AX=Y, with A belonging to various special classes like invertible, Hermitian, or unitary.

Contribution

It characterizes the existence of matrices with specific properties that solve the linear targeting problem for given data matrices X and Y.

Findings

Conditions for invertible A solving AX=Y

Existence criteria for Hermitian and positive semidefinite A

Characterization of unitary, projection, reflection, symmetric, and normal A solutions

Abstract

For given real or complex $m \times n$ data matrices $X$, $Y$, we investigate when there is a matrix $A$ such that $AX = Y$, and $A$ is invertible, Hermitian, positive (semi)definite, unitary, an orthogonal projection, a reflection, complex symmetric, or normal.