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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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