An analysis of the matrix equation AX - XB = C

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1966

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Let A, B, and C be given matrices, where A and B are of order n and m, respectively, and C is n x m . Then one can show that the matrix equation AX - XB = C always has a unique solution when A and B have no common eigen-values. If A and B have common eigenvalues, there is a solution only when C is in a special form and in that case the solution is no longer unique. Furthermore, if A = B and A has distinct eigenvalues and C is the zero matrix then X is a polynomial in A.

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