Pair Of Symmetric Matrices Research Articles

On the spectral structure of Jordan-Kronecker products of symmetric and skew-symmetric matrices

Motivated by the conjectures formulated in 2003 [28], we study interlacing properties of the eigenvalues of A⊗B+B⊗A for pairs of n-by-n matrices A,B. We prove that for every pair of symmetric matrices (and skew-symmetric matrices) with one of them at most rank two, the odd spectrum (those eigenvalues determined by skew-symmetric eigenvectors) of A⊗B+B⊗A interlaces its even spectrum (those eigenvalues determined by symmetric eigenvectors). Using this result, we also show that when n≤3, the odd spectrum of A⊗B+B⊗A interlaces its even spectrum for every pair A,B. The interlacing results also specify the structure of the eigenvectors corresponding to the extreme eigenvalues. In addition, we identify where the conjecture(s) and some interlacing properties hold for a number of structured matrices. We settle the conjectures of [28] and show they fail for some pairs of symmetric matrices A,B, when n≥4 and the ranks of A and B are at least 3.

Noisy Accelerated Power Method for Eigenproblems With Applications

This paper introduces an efficient algorithm for finding the dominant generalized eigenvectors of a pair of symmetric matrices. Combining tools from approximation theory and convex optimization, we develop a simple scalable algorithm with strong theoretical performance guarantees. More precisely, the algorithm retains the simplicity of the well-known power method but enjoys the asymptotic iteration complexity of the powerful Lanczos method. Unlike these classic techniques, our algorithm is designed to decompose the overall problem into a series of subproblems that only need to be solved approximately. The combination of good initializations, fast iterative solvers, and appropriate error control in solving the subproblems lead to a linear running time in the input sizes compared to the superlinear time for the traditional methods. The improved running time immediately offers acceleration for several applications. As an example, we demonstrate how the proposed algorithm can be used to accelerate canonical correlation analysis, which is a fundamental statistical tool for learning of a low-dimensional representation of high-dimensional objects. Numerical experiments on real-world datasets confirm that our approach yields significant improvements over the current state of the art.

Miniversal deformations of pairs of symmetric matrices under congruence

For each pair of complex symmetric matrices (A,B) we provide a normal form with a minimal number of independent parameters, to which all pairs of complex symmetric matrices (A˜,B˜), close to (A,B) can be reduced by congruence transformation that smoothly depends on the entries of A˜ and B˜. Such a normal form is called a miniversal deformation of (A,B) under congruence. A number of independent parameters in the miniversal deformation of a symmetric matrix pencil is equal to the codimension of the congruence orbit of this symmetric matrix pencil and is computed too. We also provide an upper bound on the distance from (A,B) to its miniversal deformation.

Finding Planted Subgraphs with Few Eigenvalues using the Schur--Horn Relaxation

Extracting structured subgraphs inside large graphs---often known as the planted subgraph problem---is a fundamental question that arises in a range of application domains. This problem is NP-hard in general and, as a result, significant efforts have been directed towards the development of tractable procedures that succeed on specific families of problem instances. We propose a new computationally efficient convex relaxation for solving the planted subgraph problem; our approach is based on tractable semidefinite descriptions of majorization inequalities on the spectrum of a symmetric matrix. This procedure is effective at finding planted subgraphs that consist of few distinct eigenvalues, and it generalizes previous convex relaxation techniques for finding planted cliques. Our analysis relies prominently on the notion of spectrally comonotone matrices, which are pairs of symmetric matrices that can be transformed to diagonal matrices with sorted diagonal entries upon conjugation by the same orthogonal matrix.

Determinantal representations of hyperbolic curves via polynomial homotopy continuation

A smooth curve in the real projective plane is hyperbolic if its ovals are maximally nested. By the Helton-Vinnikov Theorem, any such curve admits a definite symmetric determinantal representation. We use polynomial homotopy continuation to compute such representations numerically. Our method works by lifting paths from the space of hyperbolic polynomials to a branched cover in the space of pairs of symmetric matrices.

Symmetric matrix pencils: codimension counts and the solution of a pair of matrix equations

The set of all solutions to the homogeneous system of matrix equations (XTA +AX,XTB + BX) = (0, 0), where (A,B) is a pair of symmetric matrices of the same size, is characterized. In addition, the codimension of the orbit of (A,B) under congruence is calculated. This paper is a natural continuation of the article [A. Dmytryshyn, B. K°agstr¨om, and V.V. Sergeichuk. Skew-symmetric matrix pencils: Codimension counts and the solution of a pair of matrix equations. Linear Algebra Appl., 438:3375–3396, 2013.], where the corresponding problems for skew-symmetric matrix pencils are solved. The new results will be useful in the development of the stratification theory for orbits of symmetric matrix pencils.

On the simultaneous tridiagonalization of two symmetric matrices

We discuss congruence transformations aimed at simultaneously reducing a pair of symmetric matrices to tridiagonal–tridiagonal form under the very mild assumption that the matrix pencil is regular. We outline the general principles and propose a unified framework for the problem. This allows us to gain new insights, leading to an economical approach that only uses Gauss transformations and orthogonal Householder transformations. Numerical experiments show that the approach is numerically robust and competitive.

Linear phase FIR differentiator design based on maximum signal-to-noise ratio criterion

In this paper, a new approach to the design of digital FIR differentiators is presented. The differentiator is linear phase and has zero derivatives at DC frequency. The design is based on the maximization of signal-to-noise ratio (SNR) at the output of the differentiator. The optimal filter coefficients are obtained from the generalized eigenvector associated with the maximum eigenvalue of a pair of symmetric matrices. Estimation of the time derivative of polynomial signal, sinusoidal signal and handwritten Chinese signature is used to demonstrate that the proposed method provides better accuracy and higher SNR than the conventional differentiator methods.

Simultaneous tridiagonalization of two symmetric matrices

AbstractWe show how to simultaneously reduce a pair of symmetric matrices to tridiagonal form by congruence transformations. No assumptions are made on the non‐singularity or definiteness of the two matrices. The reduction follows a strategy similar to the one used for the tridiagonalization of a single symmetric matrix via Householder reflectors. Two algorithms are proposed, one using non‐orthogonal rank‐one modifications of the identity matrix and the other, more costly but more stable, using a combination of Householder reflectors and non‐orthogonal rank‐one modifications of the identity matrix with minimal condition numbers. Each of these tridiagonalization processes requires O(n3) arithmetic operations and respects the symmetry of the problem. We illustrate and compare the two algorithms with some numerical experiments. Copyright © 2003 John Wiley & Sons, Ltd.

Class numbers of pairs of symmetric matrices

Class numbers of pairs of symmetric matrices

An interior-point method for generalized linear-fractional programming

We develop an interior-point polynomial-time algorithm for a generalized linear-fractional problem. The latter problem can be regarded as a nonpolyhedral extension of the usual linear-fractional programming; typical example (which is of interest for control theory) is the minimization of the generalized eigenvalue of a pair of symmetric matrices linearly depending on the decision variables.

Method of centers for minimizing generalized eigenvalues

We consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices, each of which depends affinely on the decision variables. Although this problem may appear specialized, it is in fact quite general, and includes for example all linear, quadratic, and linear fractional programs. Many problems arising in control theory can be cast in this form. The problem is nondifferentiable but quasiconvex, so methods such as Kelley's cutting-plane algorithm or the ellipsoid algorithm of Shor, Nemirovsky, and Yudin are guaranteed to minimize it. In this paper we describe relevant background material and a simple interior-point method that solves such problems more efficiently. The algorithm is a variation on Huard's method of centers, using a self-concordant barrier for matrix inequalities developed by Nesterov and Nemirovsky. (Nesterov and Nemirovsky have also extended their potential reduction methods to handle the same problem.) Since the problem is quasiconvex but not convex, devising a nonheuristic stopping criterion (i.e., one that guarantees a given accuracy) is more difficult than in the convex case. We describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper, in some cases, than a more general result to Nesterov and Nemirovsky. The algorithm is demonstrated on an example: determining the quadratic Lyapunov function that optimizes a decay-rate estimate for a differential inclusion.

The GL(n)-invariant ideals of the coordinate ring of pairs of symmetric matrices with product zero

The GL(_n)-invariant ideals of the coordinate ring of pairs of symmetric matrices with product zero

Tight bounds on the spectral radius of asymmetric nonnegative matrices

For an arbitrary asymmetric nonnegative n × n matrix A we identify a pair of symmetric matrices whose largest eigenvalues bound the spectral radius of A. Furthermore, we show that these bounding matrices are best possible by characterizing matrices A which attain equality with either the upper or the lower bounding matrix. The lower bound may be extended to some matrices with negative entries provided they have no negative cycles.

A rational canonical pair form for a pair of symmetric matrices over an arbitrary field F with char F ≠ 2 and applications to finest simultaneous block diagonalizations

Kanonische Formen fur Paare symmetrischer Matrizen Uber R sind klassisch. In dieser Arbeit wird das Paarformproblem fiir syrnrnetrische Matrizenpaare Uber beliebigen Korpern F mit char F ≠2 betrachtet: Es wird bewiesen, dass ein nichtsinguliires symmetrisches Matrizenpaar S, T iiber F simultan kongruent ist zum Paar diag (Si), diag (SiQi), wobei. Qt die Jacobsonmatrix zum Elementarteiler p\' von S~*T ist und St ein Symmetrizer von Qt ist von der Form einer symmetrischen Blockmatrix, deren Blocke unterhalb der Gegendiagonalen verschwinden. Dariiberhinaus wird fiir jede Aquivalenz-klasse explizit ein Represantant angegeben. Im Anhang wird diese rationale Paarform benutzt, um die fiir R bekannten Satze iiber simultane Diagonalisier- und Blockdiagonalisierbarkeit zweier reell symmetrischer Matrizen zu erweitern auf solche fur symmstrische Matrizenpaare Uber Korpern F mit char F ≠ 2.

Uniqueness of the “nonsingular core” for Hermitian and other matrix pencils

Every pencil of hermitian matrices is conjunctive with a pencil of the form L ⊕ M, where L (the "minimal-indices" part) has no elementary divisors and M (the "nonsingular core") is a nonsingular pencil. Here it is shown that the conjunctivity type ofM is determined by that of L ⊕ M. The same method of proof applies to many other types of pencils, e.g. to congruence of pencils based on (1) a pair of symmetric matrices, (2) a pair of alternating matrices, or (3) a symmetric and an alternating matrix.

The equivalence of pairs of Hermitian matrices

Two pairs of n-ary Hermitian forms with nXn matrices A, B and C, D with elements in the complex field are equivalent if there exists a nonsingular matrix T such that T'A T = C and T'BT = D, where T' is the conjugate-transpose of T. As is usual in the study of equivalence of pairs of matrices the work divides itself into the consideration of the non-singular and singular cases. These two cases are taken up in Parts II and I respectively. In the non-singular case the rank of pA +a-B is n except for special values of p and o-. It has frequently been pointed out that in this case no generality is lost by assuming B is of rank n. In the singular case the rank r of pA +?-B is less than n for all values of p and a-, but as above no generality is lost in assuming that the rank r of B is the maximum rank of pA +?-B. By the elementary divisors of a pair of matrices A, B is meant the elementary divisors of A -NB when B is non-singular, and the elementary divisors of pA +a-B when B is singular but the determinant IpA+a-B is not identically zero in p and o-. In the non-singular case, the well known necessary and sufficient condition for the equivalence in any field of pairs of bilinear forms, or of their corresponding matrices, and for the equivalence in the field of complex numbers of pairs of symmetric matrices is that the pairs have the same elementary divisors. This condition is known to be not sufficient