Explicit solution of Sylvester and Lyapunov equations
type: Item
instance of: scholarly article
title: Explicit solution of Sylvester and Lyapunov equations
zbMATH Open document ID: 0674.15006
DOI: 10.1016/0378-4754(89)90163-8
published in: Mathematics and Computers in Simulation
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 4103263
zbMATH Keywords: Lyapunov equation
zbMATH Keywords: Sylvester matrix equation
zbMATH Keywords: Kronecker product
zbMATH Keywords: numerical example
author: Frédéric Rotella
author: Pierre Borne
MaRDI profile type: Publication
full work available at URL: 0378-4754(89)90163-8
OpenAlex ID: W2069184781
cites work: Matrix Differential Equations and Kronecker Products
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description: scientific article; zbMATH DE number 4103263