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SOLUTION OF EQUATION AX + XB = C BY INVERSION OF AN M × M OR N × N MATRIX ∗ ANTONY JAMESON† SIAM J. Appl. Math. Vol. 16, No. 5, September 1968 It is often of interest to solve the equation AX + XB = C (1) for X, where X and C are M × N real matrices, A is an M × M real matrix, and B is an N × N real matrix. A familiar example occurs in the Lyapunov theory of stability [1], [2], [3] with B = AT . Is also arises in the theory of structures [4]. Using the notation P × Q to denote the Kronecker product (Pij Q) (see [5]) in which each element of P is multipled by Q, we find that the equation written out in full for the MN unknowns x11 , x21 , . . . , x12 , . . . in terms of c11 , c21 , . . . , c12 , . . . becomes (IN × A) + B T × IM x = c, (2) where IM and IN are the M × M and N × N identity matrices. If u is a characteristic vector of A with characteristic value λ, and v is a characteristic vector of B T with characteristic value µ, then Auv T + uv T B = (λ + µ) uv T . Thus λ + µ is a characteristic value of the system (2), which can therefore be solved if and only if λi + µj 6= 0 (3) for all i, j. When A and B can both be reduced to diagonal form by similarity transformations λ1 λ2 U −1 AU = .. . λM ∗ † Received by the editors March 7, 1968. Grumman Aircraft Engineering Corporation, Bethpage, Long Island, New York 11714. 1020 and V −1 µ1 µ2 BV = .. . µN , the solution of (1) is easily obtained as X = U X̃V −1 , where x̃ij = c̃ij λi + µ j and C̃ = U −1 CV. In fact the matrix of the expanded system (2) is reduced to a diagonal form by the transformation h i T T V −1 × U (IN × A) + B T × IM V × U −1 λ1 + µ 1 λ2 + µ 1 λ3 + µ 1 .. . = . λ + µ 1 2 .. . λM + µ N It is, however, possible to obtain the solution by the inversion of an M × M or N × N matrix without recourse to diagnalization. From the expression C0 C1 C2 C3 = 0, =C = AC1 − C1 B + AC0 B = AC2 − C2 B + AC1 B = AX + XB, = A2 X − XB 2 , = A3 X + XB 3 , ··· Ck = ACk−1 − Ck−1B + ACk−2B = Ak X − (−1)k XB k , ··· Let the characteristic equations of A and B be |λI − A| = λM + a1 λM −1 + · · · + aM = (λ − λ1 ) (λ − λ2 ) · · · (λ − λM ) = 0 1021 and |λI − B| = µN + b1 µN −1 + · · · + bN = (µ − µ1 ) (µ − µ2 ) · · · (µ − µN ) = 0. According to the Cayley-Hamilton theorem these are satisfied by A and B themselves. Therefore CN − b1 CN −1 + · · · + (−1)N −1 bN −1 C1 = AN X − b1 AN −1 X + · · · + (−1)N −1 bN −1 AX + (−1)N bN X and CM − a1 CM −1 + · · · + aM −1 C1 h i = −aM X − (−1)M XB M − a1 XB M −1 + · · · + (−1)M −1 aM −1 XB . Thus h i X = G−1 CN − b1 CN −1 + · · · + (−1)N −1 bN −1 C1 = (−1)M −1 [CM + a1 CM −1 + · · · + aM −1 C1 ] H −1, where G = AN − b1 AN −1 + · · · + (−1)N bN I = (A + µ1 I) (A + µ2 I) · · · (A + µN I) and H = B M − a1 B M −1 + · · · + (−1)M aM I = (B + λ1 I) (B + λ2 I) · · · (B + λM I) . Since the determinant of a product of matrices is the product of their determinants, it is evident that G is not invertible if for any i, −µi is a charecteristic value of A, and similarly H is not invertible if for any i, −λi is a characteristic value of B; that is, G and H are not invertible if condition (3) does not hold. The coefficients a1 , a2 , · · · , aM and b1 , b2 , · · · , bN can be determined using Bocher’s identities [6]: a1 = −tr (A) , 1 a2 = − a1 tr (A) + tr A2 , 2 1 a3 = − a2 tr (A) + a1 tr A2 + tr A3 , 3 ··· 1 aM −1 tr (A) + aM −2 tr A2 + · · · + tr AM = (−1)M |A| . aM = − M 1022 In the case of the Lyapunov equation AX + XAT = C, the coefficients ai and bi coincide and could be determined in the course of taking the powers of A to form the G matrix. By adding or subtracting the characteristic equation for A, the G matrix can also be expressed in the terms of even or odd powers of A only: G = 2 (−1)N aN I + aN −2 A2 + · · · = 2 (−1)N −1 aN −1 A + aN −3 A3 + · · · . An explicit solution can be obtained by successive elimination of powers of A between the two expressions for G. For example, the solution of the Lyapunov equation for 2×2 matrices can be written as h i 1 −1 −1 T C + |A| A C A . X= 2 tr (A) The solution for 3 × 3 matrices can be written as h i 1 2 T T2 2 −1 −1 T X= . A C − ACA + CA + a2 − a1 C − a1 a3 A C A 2 (a1 a2 − a3 ) References [1] S. Lefschetz and J. P. LaSalle, Stability by Lyapunov’s Direct Method, Academic Press, New York, 1961. [2] R. A. Smith, Matrix Calculations for Liapunov quadratic forms, J. Differential Equations, 2 (1966), pp. 208-217. [3] S. Barnett and C. Storey, Analysis and synthesis of stability matrices, Ibid., 3 (1967), pp. 414-422. [4] Er-Chieh Ma, A finite series solution of the matrix equation AX − XB = C, this Journal, 14 (1966), pp. 490-495. [5] Richard Bellman, Introduction to Matrix Analysis, McGraw-Hill, New York, 1960. [6] M. Bocher, Introduction to Higher Algebra, Macmillan, New York, 1947. 1023