Let F be a field, F ¯ be an algebraic closure of it, and let n ≥ 3 be an integer. In de Seguins Pazzis (Spaces of matrices with a sole eigenvalue. Linear Multilinear Algebra. 2012;60(10):1165–1190. doi: 10.1080/03081087.2011.654118), it has been proved that ( n 2 ) + 1 is the greatest possible dimension for a linear subspace of n × n matrices with entries in F that have at most one eigenvalue in F ¯ . Also, a classification was given of those spaces that have dimension ( n 2 ) + 1 . This classification turns out to be flawed in the very special case where n = 4 and F has characteristic 2. Here, we patch the proof to obtain the correct classification. The solution depends on the structure of F as a vector space over its subfield of squares.

In this paper, we consider the regularized deteriorated positive and skew-Hermitian splitting (RDPSS) iteration method to solve singular non-Hermitian saddle point problems arising from the steady Navier–Stokes equation, which was first presented by Cao (Appl. Math. Lett., 2018). The unconditional semi-convergence property of the RDPSS iteration method and some spectral properties of the corresponding preconditioned matrix are analysed detailedly. Meanwhile, the strategies to estimate the parameters of the RDPSS preconditioner are discussed. In the end, some numerical experiments are performed to verify the feasibility and effectiveness of the proposed method and outperform several other commonly used ones.

We present a set of generators for the coordinate ring of the geometric invariant theory quotient of a variety consisting of pairs of anti-commuting matrices. For small matrix sizes, we verify that the set of generators we provide is minimal. We present an algorithm to establish the defining relations, through which we obtain all relations for matrix sizes up to five.

Compound matrices play an important role in many fields of mathematics and have recently found new applications in systems and control theory. However, the explicit formulas for these compounds are non-trivial and not always easy to use. Here, we derive new formulas for the multiplicative and additive compounds of a matrix using Kronecker products and sums. This provides a new approach to matrix compounds based on the well-known and powerful theory of Kronecker products and sums. We demonstrate several applications of these new formulas, including deriving a new expression for the additive compound of the product of two matrices.

In tensor eigenvalue problems, one is likely to be more interested in H-eigenvalues of tensors. The largest H-eigenvalue of a nonnegative tensor or of a uniform hypergraph is the spectral radius of the tensor or of the uniform hypergraph. We find upper bounds and lower bounds (interlacing inequalities) for the largest H-eigenvalue of a principal subtensor of a symmetric zero diagonal tensor that is of even order or nonnegative, as well as lower bounds for the largest H-eigenvalue of a uniform hypergraph with some vertices or edges removed. We also investigate similar problems for the least H-eigenvalues. We give examples to verify the sharpness of the bounds or in some cases for uniform hypergraphs, we characterize the equality. Particularly, for a connected linear k-uniform hypergraph G with v ∈ V ( G ) , we give a sharp lower bound for the spectral radius of G−v in terms of the spectral radius of G and the degree of v and characterize the extremal hypergraphs, and show that the maximum spectral radius of the subhypergraphs with one vertex removed is greater than or equal to the spectral radius of the hypergraph minus one, which is attained if and only if it is a Steiner system S ( 2 , k , n ) .

The aim of this paper is to consider Waring's problem for SL n ( D ) , where D is a division ring with center F such that dim F ⁡ D ≤ 4 . We show that each element of SL n ( D ) is a product of at most three square elements. As an application, let FG be a group algebra of a locally finite group G over a field F of characteristic p ≠ 2 . We show that if either p>2, or F is algebraically closed, or F is real-closed, or G is locally nilpotent, then every element in the derived subgroup ( FG ) ′ is a product of at most three squares. In this paper, we also discuss decompositions of elements into products of torsion elements by showing, for instance, that if a field F contains at least n distinct torsion elements, then every element in SL n ( F ) is a product of at most two torsion elements.

Konrad Burnik suggests a structure-preserving QR factorization for centrosymmetric matrices, known as QX factorization. In this article, we obtain the explicit expressions for rigorous perturbation bounds of the QX factorization when the original matrix is perturbed, either norm-wise or component-wise. First, using the matrix-equation approach, weak rigorous perturbation bounds are derived. Then, strong rigorous perturbation bounds are obtained by combining the modified matrix-vector equation approach with the strategy for the Lyapunov majorant function and the Banach fixed-point theorem. The mixed and component-wise condition numbers and their upper bounds are also explicitly expressed. Numerical tests illustrate the validity of the obtained results.

A new term called the strong range inclusion is introduced and its equivalent characterization is given. Based on the strong range inclusion and the generalized polar decomposition for adjointable operators on Hilbert C ∗ -modules, a weak version of the ( M , N ) -weakly complementable condition and the simplified formula for the associated bilateral shorted operator are provided, respectively. As an application, the weakly parallel summable condition for a pair of adjointable operators and their parallel sum are dealt with. Some previous results on the parallel sum are generalized and improved. Meanwhile, some new properties of the parallel sum are derived.

This paper deals with solutions of the Yang–Baxter–like operator equation AXA = XAX on the infinite dimensional Hilbert space H , where A is a rank-two bounded linear operator and X is the unknown bounded linear operator. For an operator X, necessary and sufficient conditions such that X is a solution of the equation AXA = XAX are given. With these criteria, it is in fact shown that all solutions of the nonlinear equation AXA = XAX under the corresponding conditions can be derived merely through solving some systems of linear equations.

This paper introduces the Bott-Duffin group inverse of a square matrix with respect to a subspace, which extends the Bott-Duffin inverse. We further give some of its properties, characterizations and representations by the P L -decomposition. In addition, we consider the application of the Bott-Duffin group inverse in solving a class of restricted linear equations in electrical network theory. Finally, the Bott-Duffin group inverse is applied to solve the other two classes of restricted linear equations, and the Cramer's rule for their unique solutions has also been discussed.