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Roger A. Horn

Publications and source records attributed to Roger A. Horn.

16 recordsLinked to original sources
arXiv2026

Positivity of a Hadamard Product

A notable difference between the ordinary and Hadamard products is that the Hadamard product of two singular positive semidefinite matrices can be nonsingular, and one of the factors can even be indefinite. We present an eigenvalue lower bound for a Hadamard product that depends on the rank, effective condition number, and diagonal entries of one factor, and the smallest eigenvalues of certain principal submatrices of the other factor. We give numerical examples and discuss its applications in array signal processing and matrix time series analysis.

eess.SP↗
arXiv2025

The linear targeting problem

For given real or complex $m \times n$ data matrices $X$, $Y$, we investigate when there is a matrix $A$ such that $AX = Y$, and $A$ is invertible, Hermitian, positive (semi)definite, unitary, an orthogonal projection, a reflection, complex symmetric, or normal.

math.FA↗
arXiv2017

Symplectic spaces and pairs of symmetric and nonsingular skew-symmetric matrices under congruence

Let $\mathbb F$ be a field of characteristic not $2$, and let $(A,B)$ be a pair of $n\times n$ matrices over $\mathbb F$, in which $A$ is symmetric and $B$ is skew-symmetric. A canonical form of $(A,B)$ with respect to congruence transformations $(S^TAS,S^TBS)$ was given by Sergeichuk (1988) up to classification of symmetric and Hermitian forms over finite extensions of $\mathbb F$. We obtain a simpler canonical form of $(A,B)$ if $B$ is nonsingular. Such a pair $(A,B)$ defines a quadratic form on a symplectic space, that is, on a vector space with scalar product given by a nonsingular skew-symmetric form. As an application, we obtain known canonical matrices of quadratic forms and Hamiltonian operators on real and complex symplectic spaces.

math.RT↗
arXiv2017

Specht's criterion for systems of linear mappings

W.Specht (1940) proved that two $n\times n$ complex matrices $A$ and $B$ are unitarily similar if and only if $\operatorname{trace} w(A,A^{\ast}) = \operatorname{trace} w(B,B^{\ast})$ for every word $w(x,y)$ in two noncommuting variables. We extend his criterion and its generalizations by N.A.Wiegmann (1961) and N.Jing (2015) to an arbitrary system $\mathcal A$ consisting of complex or real inner product spaces and linear mappings among them. We represent such a system by the directed graph $Q(\mathcal A)$, whose vertices are inner product spaces and arrows are linear mappings. Denote by $\widetilde Q(\mathcal A)$ the directed graph obtained by enlarging to $Q(\mathcal A)$ the adjoint linear mappings. We prove that a system $\mathcal A$ is transformed by isometries of its spaces to a system $\mathcal B$ if and only if the traces of all closed directed walks in $\widetilde Q(\mathcal A)$ and $\widetilde Q(\mathcal B)$ coincide.

math.RT↗
arXiv2013

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.

math.RT↗
arXiv2011

Simultaneous unitary equivalences

Let A, B, C, D be given finite sets of pairs of n-by-n complex matrices. We describe an algorithm to determine, with finitely many computations, whether there is a single unitary matrix U such that each pair of matrices in A is unitarily similar via U, each pair of matrices in B is unitarily congruent via U, each pair of matrices in C is unitarily similar via \bar{U}, and each pair of matrices in D is unitarily congruent via \bar{U}.

math.RT↗
arXiv2011

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.

math.RT↗
arXiv2010

Matrices that are self-congruent only via matrices of determinant one

Docovic and Szechtman, [Proc. Amer. Math. Soc. 133 (2005) 2853-2863] considered a vector space V endowed with a bilinear form. They proved that all isometries of V over a field F of characteristic not 2 have determinant 1 if and only if V has no orthogonal summands of odd dimension (the case of characteristic 2 was also considered). Their proof is based on Riehm's classification of bilinear forms. Coakley, Dopico, and Johnson [Linear Algebra Appl. 428 (2008) 796-813] gave another proof of this criterion over the fields of real and complex numbers using Thompson's canonical pairs of symmetric and skew-symmetric matrices for congruence. Let M be the matrix of the bilinear form on V. We give another proof of this criterion over F using our canonical matrices for congruence and obtain necessary and sufficient conditions involving canonical forms of M for congruence, of (M^T,M) for equivalence, and of M^{-T}M (if M is nonsingular) for similarity.

math.RT↗
arXiv2007

Canonical Forms for Unitary Congruence and *Congruence

We use methods of the general theory of congruence and *congruence for complex matrices--regularization and cosquares-to determine a unitary congruence canonical form (respectively, a unitary *congruence canonical form) for complex matrices A such that \bar{A}A (respectively, A^2) is normal. As special cases of our canonical forms, we obtain-in a coherent and systematic way-known canonical forms for conjugate normal, congruence normal, coninvolutory, involutory, projection, and unitary matrices. But we also obtain canonical forms for matrices whose squares are Hermitian or normal, and other cases that do not seem to have been investigated previously. We show that the classification problems under (a) unitary *congruence when A^3 is normal, and (b) unitary congruence when A\bar{A}A is normal, are both unitarily wild, so there is no reasonable hope that a simple solution to them can be found.

math.RT↗
arXiv2007

Canonical matrices of bilinear and sesquilinear forms

Canonical matrices are given for (a) bilinear forms over an algebraically closed or real closed field; (b) sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution; and (c) sesquilinear forms over a field F of characteristic different from 2 with involution (possibly, the identity) up to classification of Hermitian forms over finite extensions of F. A method for reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings is used to construct the canonical matrices. This method has its origins in representation theory and was devised in [V.V. Sergeichuk, Math. USSR-Izv. 31 (1988) 481-501].

math.RT↗
arXiv2007

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.

math.RT↗