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The linear targeting problem

Kyle Bierly, Stephan Ramon Garcia, Roger A. Horn

TL;DR

This work analyzes the linear targeting problem $AX=Y$ over $\mathbb{R}$ or $\mathbb{C}$ under a range of matrix structure constraints on $A$ (invertible, Hermitian, positive semidefinite, unitary, orthogonal projection, reflection, complex symmetric, and normal). It proves the fundamental null-space feasibility condition $\operatorname{null} X\subseteq\operatorname{null} Y$ and provides explicit constructiveness in each case via singular value decompositions, Moore–Penrose pseudoinverse, and block-matrix Schur-complement arguments, yielding precise existence criteria and parametrizations for $X$–to–$Y$ mappings under each structural constraint. Key results include exact solvability criteria for Hermitian, PSD, unitary, projection, and complex symmetric targeting, along with a detailed normal-targeting theorem in a special equal-dimension setting; together these give a comprehensive linear-algebraic framework for constraint-aware data mapping. The methods enable principled design of target mappings with prescribed algebraic properties, with potential applications in data alignment, signal processing, and structured linear transformations where preserving or enforcing matrix structure is essential.

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.

The linear targeting problem

TL;DR

This work analyzes the linear targeting problem over or under a range of matrix structure constraints on (invertible, Hermitian, positive semidefinite, unitary, orthogonal projection, reflection, complex symmetric, and normal). It proves the fundamental null-space feasibility condition and provides explicit constructiveness in each case via singular value decompositions, Moore–Penrose pseudoinverse, and block-matrix Schur-complement arguments, yielding precise existence criteria and parametrizations for –to– mappings under each structural constraint. Key results include exact solvability criteria for Hermitian, PSD, unitary, projection, and complex symmetric targeting, along with a detailed normal-targeting theorem in a special equal-dimension setting; together these give a comprehensive linear-algebraic framework for constraint-aware data mapping. The methods enable principled design of target mappings with prescribed algebraic properties, with potential applications in data alignment, signal processing, and structured linear transformations where preserving or enforcing matrix structure is essential.

Abstract

For given real or complex data matrices , , we investigate when there is a matrix such that , and is invertible, Hermitian, positive (semi)definite, unitary, an orthogonal projection, a reflection, complex symmetric, or normal.
Paper Structure (10 sections, 18 theorems, 64 equations)

This paper contains 10 sections, 18 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Unconstrained linear targeting
  3. Two Lemmas
  4. Hermitian targeting
  5. Positive semidefinite targeting
  6. Unitary targeting
  7. Reflection Targeting
  8. Orthogonal Projection Targeting
  9. Complex symmetric targeting
  10. Normal Targeting

Key Result

Theorem 2.1

Let $X,Y\in \mathsf{M}_{m\times n}(\mathbb{F})$ with $m\geq n\geq 1$ and $\operatorname{null} X\subseteq \operatorname{null} Y$.

Theorems & Definitions (36)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 26 more