Abstract
For given matrices A∈Fm×m, B∈Fn×n, and C∈Fm×n over an arbitrary field F, the matrix equation AX−XBT=C has a unique solution X∈Fm×n if and only if A and B have disjoint spectra. We describe an algorithm that computes the solution X for m,n⩽N with O(Nβ·logN) arithmetic operations in F, where β>2 is such that M×M matrices can be multiplied with O(Mβ) arithmetic operations, e.g., β=2.376. It seems that before no better bound than O(m3·n3) arithmetic operations was known. The state of the art in numerical analysis is O(n3+m3) flops, but these algorithms (due to Bartels/Stewart and Golub/Nash/van Loan) involve Schur decompositions, i.e., they compute the eigenvalues of at least one of A and B, and can hence not be transferred for general F.
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