The Sylvester Matrix Equation Theorem
We study the Sylvester Matrix Equation: AX-XB=C, where there matrices in the equation need not be square matrices. By thinking of this problem as a linear equation on the vector space of matrices with the same dimension as X, we can conclude by the Cayley-Hamilton Theorem ( • The Cayley-Hamilton Theorem ) that there is a solution to this equation for every matrix C if and only if the spectrums of A and B are disjoint.
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