Hermitian Positive Definite Matrix Research Articles

Indefinite QR Factorization

Let A be a Hermitian positive definite matrix given by its rectangular factor G such that A=G*G. It is well known that the Cholesky factorization of A is equivalent to the QR factorization of G. In this paper, an analogue of the QR factorization for Hermitian indefinite matrices is constructed. This problem has been considered by many authors, but the problem of zero diagonal elements has not been solved so far. Here we show how to overcome this difficulty.

Inertia theorems for pairs of matrices, II

Let L be a square matrix. A well-known theorem due to Lyapunov states that L is positive stable if and only if there exists a (Hermitian) positive definite matrix H such that LH + HL ∗ is positive definite. The main inertia theorem, due to Ostrowski, Schneider and Taussky, states that there exists a Hermitian matrix H such that LH + HL ∗ is positive definite if and only if L has no eigenvalues with zero real part; and, in that case, the inertias of L and H coincide. A pair ( A, B) of matrices of sizes p × p and p × q, respectively, is said to be positive stabilizable if there exists X such that A + BX is positive stable. In a previous paper, the results above and other inertia theorems were generalized to pairs of matrices, in order to study stabilization instead of stability. In a second paper, analogous questions about stabilization with respect to the unit disc were also considered. Denote by π( L) the number of eigenvalues of L with real positive part. In the present paper, we study the inequality π( LH + HL ∗) ⩾ l, the corresponding inequality for discrete-time systems, π( H − LHL ∗) ⩾ l, and their generalizations related with stabilization.

A generalization of T. Chan’s preconditioner

In this paper, we propose a new preconditioner for solving linear systems by the preconditioned conjugate gradient (PCG) method. The preconditioner can be thought of as a generalization of the well-known T. Chan’s preconditioner. For Hermitian positive definite matrix, we observe that our preconditioner is also Hermitian positive definite. The operation cost and convergence of the PCG method are discussed. Numerical experiments have been performed on structured problems to show the competitiveness of this preconditioner.

On a conjecture about the μ-permanent

Let A=(a ij ) be an n-by-n matrix. For any real μ, define the polynomial where ℓ (σ) is the number of inversions of the permutation σ in the symmetric group Sn . We prove that P μ (A) is a strictly increasing function of μ ∈ [−1,1], for a Hermitian positive definite nondiagonal matrix A, whose graph is a tree.

Orthogonal Hessenberg Reduction and Orthogonal Krylov Subspace Bases

We study necessary and sufficient conditions that a nonsingular matrix A can be B-orthogonally reduced to upper Hessenberg form with small bandwidth. By this we mean the existence of a decomposition AV=VH, where H is upper Hessenberg with few nonzero bands, and the columns of V are orthogonal in an inner product generated by a hermitian positive definite matrix B. The classical example for such a decomposition is the matrix tridiagonalization performed by the hermitian Lanczos algorithm, also called the orthogonal reduction to tridiagonal form. Does there exist such a decomposition when A is nonhermitian? In this paper we completely answer this question. The related (but not equivalent) question of necessary and sufficient conditions on A for the existence of short-term recurrences for computing B-orthogonal Krylov subspace bases was completely answered by the fundamental theorem of Faber and Manteuffel [SIAM J. Numer. Anal.}, 21 (1984), pp. 352--362]. We give a detailed analysis of B-normality, the central condition in both the Faber--Manteuffel theorem and our main theorem, and show how the two theorems are related. Our approach uses only elementary linear algebra tools. We thereby provide new insights into the principles behind Krylov subspace methods, that are not provided when more sophisticated tools are employed.

Estimating the Rank of the Spectral Density Matrix

Abstract. The rank of the spectral density matrix conveys relevant information in a variety of statistical modelling scenarios. This note shows how to estimate the rank of the spectral density matrix at any given frequency. The method presented is valid for any hermitian positive definite matrix estimate that has a normal asymptotic distribution with a covariance matrix the rank of which is known.

On the growth factor in Gaussian elimination for matrices with sharp angular field of values

Recently, the authors have shown that Gaussian elimination is stable for complex matrices A=B+iC where both B and C are Hermitian definite matrices. Moreover, the growth factor is less than $3\sqrt 2$ under any diagonal pivoting order. Assume now that B and C, in addition to being (positive) definite, satisfy the inequality $\displaystyle C \le \alpha B, \quad \alpha \ge 0, $ i.e., $ \displaystyle (Cx,x) \le \alpha (Bx,x) \quad \forall x \in {\bf C}^n. $ If α = 0, then A = B is a Hermitian positive definite matrix. It is well-known that, in this case, the growth factor is equal to 1. For α > 0, we establish a bound for the growth factor that has the limit 1 as α → 0.

Inertia theorems for pairs of matrices

Let L be a square matrix. A well-known theorem due to Lyapunov states that L is positive stable if and only if there exists a Hermitian positive definite matrix H such that LH+ HL * is positive definite. The main inertia theorem, due to Ostrowski, Schneider and Taussky, states that there exists a Hermitian matrix H such that LH+ HL * is positive definite if and only if L has no eigenvalues with zero real part; and, in that case, the inertias of L and H coincide. A pair ( A, B) of matrices of sizes p× p and p× q, respectively, is said to be positive stabilizable if there exists X such that A+ BX is positive stable. In this paper, we generalize Lyapunov’s theorem by giving necessary and sufficient conditions for ( A, B) being positive stabilizable. We also give generalizations of the main inertia theorem and of another inertia theorem due to Chen and Wimmer. Then we deduce a necessary condition for the existence of a Hermitian matrix H such that K:= LH+ HL * is positive semidefinite and the number of nonconstant invariant factors of xI−LK has a fixed value. This last result was inspired by another inertia theorem due to Loewy.

On determining upper bounds of maximal eigenvalue of Hermitian positive-definite matrix

We first present a new method of determining the upper bounds of maximal eigenvalue of Hermitian positive-definite matrix, which includes the Dembo's (1988) upper bound as a special case. Then we derive the kth-order upper bound /spl Lambda//sub k/ and its first-round iteration /spl Lambda//sub k//sup (1)/ which is much tighter than the Dembo's upper bound /spl Lambda//sub 0/.

Perturbation analysis of the matrix equation [formula omitted

Consider the nonlinear matrix equation X=Q+A H ( X−C) −1A, where Q is an n× n Hermitian positive definite matrix, C is an mn× mn Hermitian positive semidefinite matrix, A is an mn× n matrix, and X is the m× m block diagonal matrix defined by X= diag(X,X,…,X) , in which X is an n× n matrix. This matrix equation is connected with certain interpolation problem. In this paper, perturbation bounds and condition numbers for the maximal solution are presented, and residual bounds for an approximate solution to the maximal solution are obtained. The results are illustrated by numerical examples.

A test statistic in the complex wishart distribution and its application to change detection in polarimetric SAR data

When working with multilook fully polarimetric synthetic aperture radar (SAR) data, an appropriate way of representing the backscattered signal consists of the so-called covariance matrix. For each pixel, this is a 3/spl times/3 Hermitian positive definite matrix that follows a complex Wishart distribution. Based on this distribution, a test statistic for equality of two such matrices and an associated asymptotic probability for obtaining a smaller value of the test statistic are derived and applied successfully to change detection in polarimetric SAR data. In a case study, EMISAR L-band data from April 17, 1998 and May 20, 1998 covering agricultural fields near Foulum, Denmark are used. Multilook full covariance matrix data, azimuthal symmetric data, covariance matrix diagonal-only data, and horizontal-horizontal (HH), vertical-vertical (VV), or horizontal-vertical (HV) data alone can be used. If applied to HH, VV, or HV data alone, the derived test statistic reduces to the well-known gamma likelihood-ratio test statistic. The derived test statistic and the associated significance value can be applied as a line or edge detector in fully polarimetric SAR data also.

Inexact Rayleigh Quotient-Type Methods for Eigenvalue Computations

We consider the computation of an eigenvalue and corresponding eigenvector of a Hermitian positive definite matrix A ∈ \(\mathbb{C}^{n \times n}\), assuming that good approximations of the wanted eigenpair are already available, as may be the case in applications such as structural mechanics. We analyze efficient implementations of inexact Rayleigh quotient-type methods, which involve the approximate solution of a linear system at each iteration by means of the Conjugate Residuals method. We show that the inexact version of the classical Rayleigh quotient iteration is mathematically equivalent to a Newton approach. New insightful bounds relating the inner and outer recurrences are derived. In particular, we show that even if in the inner iterations the norm of the residual for the linear system decreases very slowly, the eigenvalue residual is reduced substantially. Based on the theoretical results, we examine stopping criteria for the inner iteration. We also discuss and motivate a preconditioning strategy for the inner iteration in order to further accelerate the convergence. Numerical experiments illustrate the analysis.

A Determinant Representation for the Distribution of Quadratic Forms in Complex Normal Vectors

Let the column vectors of X::M×N, M<N, be distributed as independent complex normal vectors with the same covariance matrix Σ. Then the usual quadratic form in the complex normal vectors is denoted by Z=XLXH where L:N×N is a positive definite hermitian matrix. This paper deals with a representation for the density function of Z in terms of a ratio of determinants. This representation also yields a compact form for the distribution of the generalized variance |Z|.

A Determinant Representation for the Distribution of a Generalised Quadratic Form in Complex Normal Vectors

Consider the quadratic form Z=YH(XLXH)−1Y where Y is a p×m complex Gaussian matrix, X is an independent p×n complex Gaussian matrix, L is a Hermitian positive definite matrix, and m⩽p⩽n. The distribution of Z has been studied for over 30 years due to its importance in certain multivariate statistics but no satisfactory numerical methods for computing this distribution appear to be available. Hence this paper deals with a representation for the density function of Z in terms of a ratio of determinants which is shown to be more amenable to numerical work than previous representations, at least for small values of p. Also for m=1 this work has applications in digital mobile radio for a specific channel where p antennas are used to receive a signal with n interferers. Some of these applications in radio communication systems are discussed.

A matrix version of the Wielandt inequality and its applications to statistics

Suppose that A is an n× n positive definite Hermitian matrix. Let X and Y be n× p and n× q matrices, respectively, such that X * Y= 0. The present article proves the following inequality, X ∗ AY Y ∗ AY − Y ∗ AX ⩽ λ 1−λ n λ 1+λ n 2 X * AX , where λ 1 and λ n are respectively the largest and smallest eigenvalues of A, and M − stands for a generalized inverse of M. This inequality is an extension of the well-known Wielandt inequality in which both X and Y are vectors. The inequality is utilized to obtain some interesting inequalities about covariance matrix and various correlation coefficients including the canonical correlation, multiple and simple correlations. Some applications in parameter estimation are also given.

A relative perturbation bound for positive definite matrices

We give a sharp estimate for the eigenvectors of a positive definite Hermitian matrix under a floating-point perturbation. The proof is elementary.

A relative perturbation bound for positive definite matrices

We give a sharp estimate for the eigenvectors of a positive definite Hermitian matrix under a floating-point perturbation. The proof is elementary.

An inequality for positive definite matrices with applications to combinatorial matrices

If A ∈ M n( C) is a positive definite Hermitian matrix, d the average of the diagonal entries of A, and f the average of the absolute values of the off-diagonal entries of A, then det A ⩽ ( d − f) n−1 [ d + ( n − 1) f]. As a corollary we obtain a strengthening of Hadamard's inequality for positive definite matrices. The results can be used to prove inequalities for the determinants of (± 1) matrices, (0, 1) matrices, positive matrices, stochastic matrices, and constant-column-sum matrices.

On the convergence of the MAOR method

In order to solve a linear system Ax = b, Hadjidimos et al. (1992) defined a class of modified AOR (MAOR) method, whose special case implies the MSOR method. In this paper, some sufficient and/or necessary conditions for convergence of the MAOR and MSOR methods will be achieved, when A is a two-cyclic matrix and when A is a Hermitian positive-definite matrix, an H-, L- or M-matrix, and a strictly or irreducibly diagonally dominant matrix. The convergence results on the MSOR method are better than some known theorems. The optimum parameters and the optimum spectral radii of the MAOR and MSOR methods are obtained, which also answers the open problem given by Hadjidimos et al.

A new FIR system identification method based on fourth-order cumulants: Application to blind equalization

Recently, numerous Finite Impulse Response (FIR) system identification methods based on High Order Statistics (HOS) have been proposed in the literature. These methods can be classified into three categories: closed-form solutions, linear algebra solutions and nonlinear optimization based solutions. Because of their simplicity, closed-form and linear algebra solutions are the ones most used in practice. In this paper, we propose a new explicit solution based on fourth-order cumulants. Compared to existing closed-form solutions, the proposed solution uses more statistics and compared to linear algebra solutions it needs less arithmetical operations. This solution is based on a Cholesky type decomposition of a positive definite Hermitian matrix made up of fourth-order cumulants. An efficient algorithm is given to estimate this matrix. Then, the proposed FIR system identification method is applied to estimate the impulse response (i.r.) coefficients of a radio communication channel in order to “open the eyes” of the channel by minimizing the Mean-Square Error (MSE) criterion. The performance of the proposed method is illustrated by some simulation results.