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Given the matrix equation A X + X B + f (X) C = D in the unknown n × m matrix X , we analyze existence and uniqueness conditions, together with computational solution strategies for f : R n × m → R being a linear or nonlinear function. We characterize different properties of the matrix equation and of its solution, depending on the considered classes of functions f. Our analysis mainly concerns small dimensional problems, though several considerations also apply to large scale matrix equations. [ABSTRACT FROM AUTHOR]

Let A be a unital complex semisimple Banach algebra, and M A denote its maximal ideal space. For a matrix M ∈ A n × n , M ˆ denotes the matrix obtained by taking entry-wise Gelfand transforms. For a matrix M ∈ C n × n , σ (M) ⊂ C denotes the set of eigenvalues of M. It is shown that if A ∈ A n × n and B ∈ A m × m are such that for all φ ∈ M A , σ (A ˆ (φ)) ∩ σ (B ˆ (φ)) = ∅ , then for all C ∈ A n × m , the Sylvester equation A X − X B = C has a unique solution X ∈ A n × m. As an application, Roth's removal rule is proved in the context of matrices over a Banach algebra. [ABSTRACT FROM AUTHOR]

This paper introduces a novel framework for the solution of (large-scale) Lyapunov and Sylvester equations derived from numerical integration methods. Suitable systems of ordinary differential equations are introduced. Low rank approximations of their solutions are produced by Runge-Kutta methods. Appropriate Runge-Kutta methods are identified following the idea of geometric numerical integration to preserve a geometric property, namely a low rank residual. For both types of equations we prove the equivalence of one particular instance of the resulting algorithm to the well known ADI iteration. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study some features of slice semi-regular functions SEM (Ω) on a circular domain Ω contained in the skew-symmetric algebra of quaternions H via the analysis of a family of linear operators built from left and right ⁎-multiplication on SEM (Ω) ; this class of operators includes the family of Sylvester-type operators S f , g. Our goal is achieved by a strategy based on a matrix interpretation of these operators as we show that SEM (Ω) can be seen as a 4-dimensional vector space on the field SEM R (Ω). We then study the rank of S f , g and describe its kernel and image when it is not invertible, finding meaningful differences in the cases when the rank is either 2 or 3. By using these results, we are able to characterize when the functions f and g are either equivalent under ⁎-conjugation or intertwined by means of a zero divisor, thus proving a number of statements on the behaviour of slice semi-regular functions. In this way, information about the operator obtained by linear algebra techniques give as a significant application the solution of a problem in an area of function theory which had a remarkable development in the last decade (see [16]). We also provide a complete classification of idempotents and zero divisors on product domains of H. [ABSTRACT FROM AUTHOR]

We provide an amended version of Corollaries 7 and 9 in [De Terán, Iannazzo, Poloni, Robol, “Solvability and uniqueness criteria for generalized Sylvester-type equations”]. These results characterize the unique solvability of the matrix equation A X B + C X ⋆ D = E (where the coefficients need not be square) in terms of an equivalent condition on the spectrum of certain matrix pencils of the same size as one of its coefficients. [ABSTRACT FROM AUTHOR]

We give a new characterization of the solution set of non-symmetric algebraic Riccati equations involving real matrices. Our characterization involves the use of invariant subspaces of the coefficient matrices. We also give a poset structure on the solutions of ARE and explore some properties of this poset (partially ordered set). [ABSTRACT FROM AUTHOR]

We present necessary and sufficient conditions for the existence of a unique solution of the generalized ⋆-Sylvester matrix equation A X B + C X ⋆ D = E , where A , B , C , D , E are square matrices of the same size with real or complex entries, and where ⋆ stands for either the transpose or the conjugate transpose. This generalizes several previous uniqueness results for specific equations like the ⋆-Sylvester or the ⋆-Stein equations. [ABSTRACT FROM AUTHOR]

This work concerns the distance in the 2-norm from a given matrix polynomial to a nearest polynomial with a specified number of its eigenvalues at specified locations in the complex plane. Initially, we consider perturbations of the constant coefficient matrix only. A singular value optimization characterization is derived for the associated distance. We also consider the distance in the general setting, when all of the coefficient matrices are perturbed. In this general setting, we obtain a lower bound in terms of another singular value optimization problem. The singular value optimization problems derived facilitate the numerical computation of the distances. [ABSTRACT FROM AUTHOR]

Abstract: Fast algorithms for enclosing solutions of Sylvester equations are proposed. The results obtained by these algorithms are “verified” in the sense that all the possible rounding errors have been taken into account. For developing these algorithms, theories which directly supply error bounds for numerical solutions are established. The proposed algorithms require only operations if A and B are diagonalizable. Techniques for accelerating the enclosure and obtaining smaller error bounds are introduced. Numerical results show the properties of the proposed algorithms. [Copyright &y& Elsevier]

Abstract: Given matrices A i , B i and C i of corresponding dimensions over a field , we prove that: (i) if are simultaneously similar to , then there exists a simultaneous solution X to the matrix Sylvester equations ; and (ii) if are simultaneously equivalent to then there exist simultaneous solutions to the matrix equations . We also show that analogous results hold for mixed pairs of matrix Sylvester equations , and for generalized Stein equations . [Copyright &y& Elsevier]

Abstract: We investigate simultaneous solutions of the matrix Sylvester equations , where and are k-tuples of commuting matrices of order and , respectively. We show that the matrix Sylvester equations have a unique solution X for every compatible k-tuple of matrices if and only if the joint spectra and are disjoint. We discuss the connection between the simultaneous solutions of Sylvester equations and related questions about idempotent matrices separating disjoint subsets of the joint spectrum, spectral mapping for the differences of commuting k-tuples, and a characterization of the joint spectrum via simultaneous solutions of systems of linear equations. [Copyright &y& Elsevier]

Abstract: In this paper we consider the special Sylvester equation for fixed matrices M and N, where a positive definite solution X is sought. We show that the solution sets varying over provide a new family of geodesic submanifolds in the symmetric Riemannian manifold of positive definite matrices which is stable under congruence transformations; it consists of geodesically complete convex cones of invariant under Cartan symmetries. It is further shown that the solution set is stable under the iterative means obtained by the weighted arithmetic, harmonic and geometric means. [Copyright &y& Elsevier]

Abstract: It is well known that the Sylvester matrix equation AX + XB = C has a unique solution X if and only if 0∉spec(A)+spec(B). The main result of the present article are explicit formulas for the determinant of X in the case that C is one-dimensional. For diagonal matrices A, B, we reobtain a classical result by Cauchy as a special case. The formulas we obtain are a cornerstone in the asymptotic classification of multiple pole solutions to integrable systems like the sine-Gordon equation and the Toda lattice. We will provide a concise introduction to the background from soliton theory, an operator theoretic approach originating from work of Marchenko and Carl, and discuss examples for the application of the main results. [Copyright &y& Elsevier]

Abstract: The theme of the present paper is to introduce and study two different versions of tensor products of functional models, one over the underlying field and the other over the corresponding algebra of polynomials, as well as related models based on the Kronecker product of polynomial matrices. In the process, we study the Sylvester equation and its reduction to a polynomial matrix equation. We analyse the relation between the two tensor products and use this to elucidate the role of the Anderson-Jury generalized Bezoutians in this context. [Copyright &y& Elsevier]

Abstract: We give new lower bounds on the Rayleigh–Ritz approximations of a part of the spectrum of an elliptic operator. Furthermore, we present bounds for the accompanying Ritz vectors. The bounds include a form of a relative gap between the Ritz values and the rest of the spectrum of the operator. A model example shows that the obtained bounds may be very sharp. [Copyright &y& Elsevier]

Abstract: We formulate and provide a solution to an approximation problem that occurs in various settings: Finding an optimal additive decomposition of a given Hermitian Hilbert–Schmidt operator, in a term commuting with a second Hermitian compact operator and a term as small as possible in the trace norm sense. In the finite-dimensional case, we show how to interpret our result through a Sylvester equation. An application to a quantum information problem and an interpretation in quantum probability are also sketched. [Copyright &y& Elsevier]