# Eikonal Blog

## 2010.07.02

### Sylvester equation etc

Filed under: mathematics — Tags: — sandokan65 @ 10:13

Definition: $AY+YB=X$ for $A, B, X, Y \in {\Bbb C}^{n\times n}$

For $n=2$ with $A^2 = - \delta_A {\bf 1}$ and $B^2 = - \delta_B {\bf 1}$ (i.e. with $tr(A)=tr(B)=0$), one has unique solution:

$Y = \frac1{\delta_B-\delta_A}(A X - X B)$

the existence condition here is that $\delta_B \ne \delta_A$ i.e. $\det(A) \ne \det(B)$

For $B=\mu A$ i.e. $AY+\mu YA=X$ we get:

$Y = \frac1{-\delta_A (1-\mu^2)}(A X - \mu X A)$,
with the condition of existence $\mu\ne1$.

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