Representations of quivers and mixed graphs

This is a survey article for "Handbook of Linear Algebra", 2nd ed., Chapman & Hall/CRC, 2014. An informal introduction to representations of quivers and finite dimensional algebras from a linear algebraist's point of view is given. The notion of quiver representations is extended to representations of mixed graphs, which permits one to study systems of linear mappings and bilinear or sesquilinear forms. The problem of classifying such systems is reduced to the problem of classifying systems of linear mappings

Normal matrices with a dominant eigenvalue and an eigenvector with no zero entries

AbstractWe say that a square complex matrix is dominant if it has an algebraically simple eigenvalue whose modulus is strictly greater than the modulus of any other eigenvalue; such an eigenvalue and any associated eigenvector are also said to be dominant. We explore inequalities that are sufficient to ensure that a normal matrix is dominant and has a dominant eigenvector with no zero entries. For a real symmetric matrix, these inequalities force the entries of a dominant real eigenvector to have a prescribed sign pattern. In the cases of equality in our inequalities, we find that exceptional extremal matrices must have a very special form

A regularization algorithm for matrices of bilinear and sesquilinear forms

We give an algorithm that uses only unitary transformations and for each square complex matrix constructs a *congruent matrix that is a direct sum of a nonsingular matrix and singular Jordan blocks.Comment: 18 page

Minkowski sums and Hadamard products of algebraic varieties

We study Minkowski sums and Hadamard products of algebraic varieties. Specifically we explore when these are varieties and examine their properties in terms of those of the original varieties.Comment: 25 pages, 7 figure

Canonical forms for complex matrix congruence and *congruence

Canonical forms for congruence and *congruence of square complex matrices were given by Horn and Sergeichuk in [Linear Algebra Appl. 389 (2004) 347-353], based on Sergeichuk's paper [Math. USSR, Izvestiya 31 (3) (1988) 481-501], which employed the theory of representations of quivers with involution. We use standard methods of matrix analysis to prove directly that these forms are canonical. Our proof provides explicit algorithms to compute all the blocks and parameters in the canonical forms. We use these forms to derive canonical pairs for simultaneous congruence of pairs of complex symmetric and skew-symmetric matrices as well as canonical forms for simultaneous *congruence of pairs of complex Hermitian matrices.Comment: 31 page

Bounds on the spectral radius of a Hadamard product of nonnegative or positive semidefininte matrices

X. Zhan has conjectured that the spectral radius of the Hadamard product of two square nonnegative matrices is not greater than the spectral radius of their ordinary product. We prove Zhan’s conjecture, and a related inequality for positive semidefinite matrices, using standard facts about principal submatrices, Kronecker products, and the spectral radiu

Inequalities for C-S seminorms and Lieb functions

AbstractLet Mn be the space of n × n complex matrices. A seminorm ‖ · ‖ on Mn is said to be a C-S seminorm if ‖A*A‖ = ‖AA*‖ for all A ∈ Mn and ‖A‖≤‖B‖ whenever A, B, and B-A are positive semidefinite. If ‖ · ‖ is any nontrivial C-S seminorm on Mn, we show that ‖∣A‖∣ is a unitarily invariant norm on Mn, which permits many known inequalities for unitarily invariant norms to be generalized to the setting of C-S seminorms. We prove a new inequality for C-S seminorms that includes as special cases inequalities of Bhatia et al., for unitarily invariant norms. Finally, we observe that every C-S seminorm belongs to the larger class of Lieb functions, and we prove some new inequalities for this larger class

A canonical form for nonderogatory matrices under unitary similarity

A square matrix is nonderogatory if its Jordan blocks have distinct eigenvalues. We give canonical forms (i) for nonderogatory complex matrices up to unitary similarity and (ii) for pairs of complex matrices up to similarity, in which one matrix has distinct eigenvalues. The types of these canonical forms are given by undirected and, respectively, directed graphs with no undirected cycles.Comment: 18 page