Given A Rmm, B Rnn, C Rmn, the equation AX XB = C (3) is known as Sylvesters equation
Chapter 6, Problem 29(choose chapter or problem)
Given A Rmm, B Rnn, C Rmn, the equation AX XB = C (3) is known as Sylvesters equation. An m n matrix X is said to be a solution if it satisfies (3). (a) Show that if B has Schur decomposition B = UTUH , then Sylvesters equation can be transformed into an equation of the form AY YT = G, where Y = XU and G = CU. (b) Show that (A t11 I )y1 = g1 (A t j j I )yj = gj + _j1 i=1 ti jyj , j = 2, . . . , n (c) Show that if A and B have no common eigenvalues, then Sylvesters equation has a solution.
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