Solving the generalized Sylvester matrix equation AV + BW = V F via Kronecker map

Abstract

This note considers the solution to the generalized Sylvester matrix equation AV + BW = V F with F being an arbitrary matrix, where V and W are the matrices to be determined. With the help of the Kronecker map, an explicit parametric solution to this matrix equation is established. The proposed solution possesses a very simple and neat form, and allows the matrix F to be undetermined.

Explicit and Iterative Methods for Solving the Matrix Equation AV + BW = EVF + C

“…Lemma 8 [10]. Let A, E 2ℝ n × n , B 2ℝ n × r and F 2ℝ p × p , then the matrix equation (1) has rp degree of freedom iff rank[A À sE B] = n , ∀ s 2 σ(F).…”

Real-time covariance estimation for the local level model

“…Now define with and then substitute into (5), i.e. Note that the cases and , are embedded in . In the general case, we can obtain the solution of (5) numerically, by considering it as a special case of generalized Sylvester matrix equations (Wu et al. , 2008).…”

Lanczos version of BCR algorithm for solving the generalised second‐order Sylvester matrix equation

“…Zhou et al analysed the computational complexity of the Smith iteration and its variations for solving the Stein matrix equation [24] In [25–27], the complete general parametric expressions for the solution pair of the generalised Sylvester matrix equation were introduced. With the help of the Kronecker map and by applying the Sylvester sum as tools, an explicit parametric solution pair to the generalised Sylvester matrix equation (5) was proposed in [28]. Duan in [29] provides a complete general parametric solution of the second‐order Sylvester matrix equation (1).…”