On the inverse of the Hadamard product of a full rank matrix and an angle matrix

TL;DR

This paper addresses the problem of inverting the Hadamard product of a full-rank matrix and an angle matrix. It defines the angle matrix ${\mathbf{\Theta}}={\mathbf{v}}{\mathbf{u}}^T$ with ${\mathbf{v}}=(e^{j\theta_i})$ and ${\mathbf{u}}=(e^{j\phi_i})$, proves ${\mathbf{\Theta}}^{\circ(-T)}={\mathbf{\Theta}}^H$, and uses the identity $|{\mathbf{A}}\circ{\mathbf{\Theta}}|=|{\mathbf{A}}|e^{j\sum_i(\theta_i+\phi_i)}$ to derive inverse forms. The main contributions are closed-form Moore–Penrose inverses for ${\mathbf{A}}\circ{\mathbf{\Theta}}$ with full-rank ${\mathbf{A}}$, yielding exact inverses $( {\mathbf{A}}\circ{\mathbf{\Theta}})^{-1}={\mathbf{A}}^{-1}\circ{\mathbf{\Theta}}^H$ in the square case and rank-dependent pseudoinverses in non-square cases; the results rely on the product identity $( {\mathbf{A}}\circ{\mathbf{\Theta}})^H( {\mathbf{A}}\circ{\mathbf{\Theta}})=( {\mathbf{A}}^H{\mathbf{A}})\circ (\tfrac{1}{m}{\mathbf{\Theta}}^H{\mathbf{\Theta}})$ and standard pseudoinverse theory, contributing to the algebraic understanding of Hadamard products.

Abstract

By the definition of an angle matrix, we investigate the inverse of the Hadamard product of a full rank and an angle matrices. Our proof involves standard matrix analysis. It enriches the algebra of Hadamard products.

Theorems & Definitions (13)

• Definition 1
• Definition 2
• Lemma 1
• Lemma 2
• Theorem 3
• proof
• corollary 4
• proof
• Lemma 5
• proof