---
tag: 312GG23DN4
kind: definition
name: Sylvester equation
msc: [15A24, 65F45]
defines: M9SW4NVP2H
uses: [0VD9ZSWWJG, 04C5ZBG1WT, 0BQAA2BK3Q, 0N7ENQRCXN]
sources:
  - url: https://en.wikipedia.org/wiki/Sylvester_equation
---

## Definition

Let $\mathbb{K}$ be a [[field|04C5ZBG1WT]] and let $m, n \ge 1$.
Let $A$ be an $m \times m$ [[matrix|0VD9ZSWWJG]], $B$ an $n \times n$ matrix and $C$ an $m \times n$ matrix over $\mathbb{K}$.
The *Sylvester equation* for $A$, $B$, $C$ is the equation
$$AX + XB = C$$
for an unknown $m \times n$ matrix $X$ over $\mathbb{K}$.

## Remarks

The map $X \mapsto AX + XB$ is linear, and with the [[Kronecker product|0N7ENQRCXN]] the equation reads $(I_n \otimes A + B^T \otimes I_m) \operatorname{vec}(X) = \operatorname{vec}(C)$.
It has a unique solution for every $C$ if and only if $A$ and $-B$ have no common [[eigenvalue|0BQAA2BK3Q]] in an algebraic closure of $\mathbb{K}$.
Example: $A = \operatorname{diag}(1, 2)$, $B = 1$ and $C = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ give $x_1 + x_1 = 2$ and $2x_2 + x_2 = 3$, so $X = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$.
Non-example: the equation $AXB = C$ is not a Sylvester equation, since $X$ is multiplied on both sides in one term.
For $B = -A$, the equation $AX - XA = C$ never has a unique solution, because $A$ and $A$ share their eigenvalues.