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A proof of Sylvester's theorem

Saptak Bhattacharya

Abstract

We give a new elementary proof of existence and uniqueness of a solution to the Sylvester equation $AX-XB=Y$

A proof of Sylvester's theorem

Abstract

We give a new elementary proof of existence and uniqueness of a solution to the Sylvester equation
Paper Structure (2 sections, 1 theorem, 13 equations)

This paper contains 2 sections, 1 theorem, 13 equations.

Table of Contents

  1. Introduction
  2. Main result

Key Result

Theorem 1

Let $\mathcal{H}$ and $\mathcal{K}$ be finite dimensional Hilbert spaces and let $A\in \mathcal{L}(\mathcal{K})$ and $B\in\mathcal{L}(\mathcal{H})$ with $\sigma(A)\cap\sigma(B)=\emptyset$. Then for every $Y\in \mathcal{L}(\mathcal{H},\mathcal{K})$ there exists a unique $X\in\mathcal{L}(\mathcal{H},\

Theorems & Definitions (2)

  • Theorem 1
  • proof