Explicit solution of Sylvester and Lyapunov equations
Consider a Sylvester matrix equation (1) \(AX+XB=C\) or a discrete Sylvester equation (2) \(X+LXB=C\) where \(A\),\(B\),\(C\),\(L\) are constant \((p\times p) \), \(q\times q)\), \((p\times q)\), \((p\times p)\) dimensional matrices and \(X\) is the \((p\times q)\) unknown matrix. This paper shows that the exact solution of (1) or (2) can be calculated by the inversion of a \((\min(p,q)\times \min(p,q))\) matrix. For a \((j\times k)\) matrix \(M=(M_1,\ldots,M_ k)\), \(\text{vec}(M)\) is defined as \[ \text{vec}(M)=\left[\begin{matrix} M_1 \\ \vdots \\ M_ k \end{matrix} \right]. \] Let \(M\times N\) denote the Kronecker product of the matrices \(M\) and \(N\). The following theorems are the main results of this paper: Theorem 4: If \(A\) and \(-B\) have distinct eigenvalues, the solution of (1) is giv en by: \[ \text{vec}(X) = -\left[\sum^{q-1}_{i=0} B_ i\otimes(-A)^ i\right] \left[I_ q\otimes\left((-A)^ q + \sum^{q-1}_{i=0} b_ i(-A)^ i\right) ^{-1}\right] \text{vec}(C) \] where the \(b_ i\) are the coefficients of the characteristic polynomial of \(B\) and: \(B_{q-1}=I_ 1\), \(i\in \{0,\ldots,q-2\}\), \(B_ i=B^ T B_{i+1} + b_{i+1} I_ q\). Theorem 6: If the eigenvalues of \(B\) are distinct of the inverse of the eigenvalues of \(L\), the solution of the discrete Sylvester equation (2) is given by: \[ \text{vec}(X) = -\left[\sum^{q-1}_{i=0} (-1)^ i (B_ i\otimes L^{q-1-i})\right] \left[I_ q\otimes \left\{(-I_ p)^ q + \sum^{q-1}_{i=0} b_ i L^{q-i } (-I_ p)^ i\right\}^{-1}\right] \text{vec}(C) \] where the \(b_ i\) and \(B_ i\) are defined in the same manner as in Theorem 4. A numerical example is given.
scientific article; zbMATH DE number 4103263