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Let A = {Aij}i,j∈I, where I is an index set, be a doubly indexed family of matrices, where Aij is ni × nj. For each i∈I, let Vi be an ni-dimensional vector space. We say A is reducible in the coupled sense if there exist subspaces, Ui ⊆ Vi, with Ui ≠{0} for at least one i∈I, and Ui ≠ Vi for at least one i, such that Aij(Uj) ⊆ Ui for all i, j. Let B = {Bij}i,j∈I also be a doubly indexed family of matrices, where Bij is mi times mj. For each i∈I, let Xi be a matrix of size ni times mi. Suppose Aij} Xj = Xi Bij for all i, j. We prove versions of Schur's lemma for A, mathcal B satisfying coupled irreducibility conditions. We also consider a refinement of Schur's lemma for sets of normal matrices and prove corresponding versions for A, mathcal B satisfying coupled normality and coupled irreducibility conditions. [ABSTRACT FROM AUTHOR]

We consider the problem of efficiently solving Sylvester and Lyapunov equations of medium and large scale, in case of rank-structured data, i.e., when the coefficient matrices and the right-hand side have low-rank off-diagonal blocks. This comprises problems with banded data, recently studied in [A. Haber and M. Verhaegen, Automática J. IFAC, 73 (2016), pp. 256--268; D. Palitta and V. Simoncini, Numerical Methods for Large-Scale Lyapunov Equations with Symmetric Banded Data, preprint, arxiv, 1711.04187, 2017], which often arise in the discretization of elliptic PDEs. We show that, under suitable assumptions, the quasiseparable structure is guaranteed to be numerically present in the solution, and explicit novel estimates of the numerical rank of the off-diagonal blocks are provided. Efficient solution schemes that rely on the technology of hierarchical matrices are described, and several numerical experiments confirm the applicability and efficiency of the approaches. We develop a MATLAB toolbox that allows easy replication of the experiments and a ready-to-use interface for the solvers. The performances of the different approaches are compared, and we show that the new methods described are efficient on several classes of relevant problems. [ABSTRACT FROM AUTHOR]

We prove Roth-type theorems for systems of matrix equations including an arbitrary mix of Sylvester and *-Sylvester equations, in which the transpose or conjugate transpose of the unknown matrices also appear. In full generality, we derive consistency conditions by proving that such a system has a solution if and only if the associated set of 2 × 2 block matrix representations of the equations are block diagonalizable by (linked) equivalence transformations. Various applications leading to several particular cases have already been investigated in the literature, some recently and some long ago. Solvability of these cases follow immediately from our general consistency theory. We also show how to apply our main result to systems of Stein-type matrix equations. [ABSTRACT FROM AUTHOR]

We consider the Sylvester equation AX -XB+C = 0, where the matrix C ϶ Rn×m is of low rank and the spectra of A ϶ Rn×n and B ϶ Rm×m are separated by a line. The solution X can be approximated in a data-sparse format, and we develop a multigrid algorithm that computes the solution in this format. For the multigrid method to work, we need a hierarchy of discretizations. Here the matrices A and B each stem from the discretization of a partial differential operator of elliptic type. The algorithm is of complexity O(n+m), or, more precisely, if the solution can be represented with (n + m)k data (k ∼ log(n + m)), then the complexity of the algorithm is O((n + m)k2). [ABSTRACT FROM AUTHOR]

The distance to uncontrollability for a linear control system is the distance (in the 2-norm) to the nearest uncontrollable system. We present an algorithm based on methods of Gu and Burke-Lewis-Overton that estimates the distance to uncontrollability to any prescribed accuracy. The new method requires O(n4) operations on average, which is an improvement over previous methods which have complexity O(n6), where n is the order of the system. Numerical experiments indicate that the new method is reliable in practice. [ABSTRACT FROM AUTHOR]

We investigate the behavior of isotropic invariant subspaces of skew-Hamiltonian matrices under structured perturbations. It is shown that finding a nearby subspace is equivalent to solving a certain quadratic matrix equation. This connection is used to derive meaningful error bounds and condition numbers that can be used to judge the quality of invariant subspaces computed by strongly backward stable eigensolvers. [ABSTRACT FROM AUTHOR]