On properties of Sylvester and Lyapunov operators

Abstract

Sylvester and Lyapunov operators in real and complex matrix spaces are studied, which include as particular cases the operators arising in the theory of linear time-invariant systems. Let M: F m×n→ F p×q be a linear operator, where F= R or F= C . The operator M is elementary if there exist matrices A∈ F p×m and B∈ F q×n , such that M[X]=AXB . Each M can be represented as a sum of minimum number of elementary operators, called the Sylvester index of M . An expression for the Sylvester index of a general linear operator M is given. An important tool here is a special permutation operator V p,m: F pq×mn→ F pm×nq such that the image V p,m(B T ⊗A) of the matrix of a non-zero elementary operator is equal to the rank 1 matrix vec[A] row[B] , where vec[X] and row[X] are the column-wise and row-wise vector representation of the matrix X. The application of V p,m reduces a sum of Kronecker products of matrices to the standard product of two matrices. A linear operator L: F n×n→ F n×n is a Lyapunov operator if ( L[X]) *= L[X *] , where the star denotes transposition in the real case and complex conjugate transposition in the complex case. Characterisations and parametrisations of the sets of real and complex Lyapunov operators are given and their dimensions are found. Relevant Lyapunov indexes for Lyapunov operators are introduced and calculated. Similar results are given also for several classes of Lyapunov-like linear and pseudo-linear operators. The concept of Lyapunov singular values of a Lyapunov operator is introduced and the application of these values to the sensitivity and a posteriori error analysis of Lyapunov equations is discussed.