Forward stable eigenvalue decomposition of rank-one modifications of diagonal matrices

The inverse eigenvalue problems for symmetric Toeplitz matrices with complex-valued (IEPSCTM) and real-valued (IEPSRTM) entries are studied. The main tools are complex and real algebraic geometry. In the complex case it is shown that the IEPSCTM is solvable for most spectra and always solvable for $n\leqq 4$. In the real case the natural decomposition of the space of all $n \times n$ real symmetric Toeplitz matrices to a finite number of connected components of matrices with a simple spectrum is given. It is then shown that the solvability of the IEPSRTM for all spectra can be deduced if the corresponding map to the IEPSRTM has a nonzero degree for at least one component. This is the case for $n\leqq 4$, which gives an alternative proof to Delsarte and Genin’s results. The IEPSRTM for odd Toeplitz matrices with real-valued entries is also considered.

The inverse eigenvalue problem for symmetric matrices (IEP) is formulated as a system of two matrix equations. For solving the system a variation of Newton's method is used which has been proposed by Fusco and Zecca [Calcolo XXIII (1986), pp. 285–303] for the simultaneous computation of eigenvalues and eigenvectors of a given symmetric matrix. an iteration step of this method consists of a Newton step followed by an orthonormalization with the consequence that each iterate satisfies one of the given equations. The method converges locally y quadratically to regular solutions. The algorithm and some numerical examples are presented. In addition, it is shown that the so called Method III proposed by Friedland, Nocedal, and Overton [SIAM J. Numer. Anal.,24 (1987), pp. 634–667] for solving IEP may be constructed similarly to the method presented here

In this note we review the technique for rational Chebyshev approximation as given by the generalization of the Remes algorithm and point out its connection to the generalized eigenvalue problem for symmetric Hankel matrices. By a classical example and some recent work it can be shown that interpolation with rational functions (even) on equidistant knots may lead to results that are superior to Chebyshev approximation by means of polynomials for comparable numerical work.

In this paper, the symmetric positive semidefinite Procrustes problem is considered. By using matrix inner product and matrix decomposition theory, an explicit expression of the solution is given. Based on the explicit expression given in this paper, it is easy to derive the explicit expression of the solution given by Woodgate [K.G. Woodgate. Least-squares solution of F = PG over positive semidefinite symmetric P . Linear Algebra Appl., 245:171–190, 1996.] and by Liao [A.P. Liao. On the least squares problem of a matrix equation. J. Comput. Math., 17:589–594, 1999.] for the Procrustes problem in some special cases. The explicit expression given in this paper also shows that the solution of the special inverse eigenvalue problem considered by Zhang [L. Zhang. A class of inverse eigenvalue problem for symmetric nonnegative definite matrices. J. Hunan Educational Inst., 2:11–17, 1995 (in Chinese).] can be computed exactly. Examples to illustrate the correctness of the theory results are given.

20aEA-10 Cavity analysis of the first eigenvalue problem of large symmetric random sparse matrices

We consider numerical methods for the extreme eigenvalue problem of large scale symmetric positive definite matrices. By the variational principle, the extreme eigenvalue can be obtained by minimizing some unconstrained optimization problem. Firstly, we propose two adaptive nonmonotone Barzilai-Borwein-like methods for the unconstrained optimization problem. Secondly, we prove the global convergence of the two algorithms under some conditions. Thirdly, we compare our methods with eigsand the power method for the standard test problems from the UF Sparse Matrix Collection. The primary numerical experiments indicate that the two algorithms are promising.

In paper the generalized real eigenvalue and fuzzy eigenvector of a crisp real symmetric matrix with respect to another real symmetric matrix is studied. The original generalized fuzzy eigen problem is extended into a crisp generalized eigen problem of a real symmetric matrix with high orders using the arithmetic operation of LR fuzzy matrix and vector. Two cases are analysed: (a) the unknown eigenvalue λ is a non negative real number; (b) the unknown eigenvalue λ is a negative real number. Two computing models are established and an algorithm for finding the generalized fuzzy eigenvector of a real symmetric matrix is derived. Moreover, a sufficient condition for the existence of a strong generalized fuzzy eigenvector is given. Some numerical examples are shown to illustrated our proposed method.

Componentwise analysis of direct factorization of real symmetric and hermitian matrices

The set of Fortran subroutines discussed an implementation of the algorithm for finding the eigenvectors, x, and eigenvalues, lambda, such that Ax = lambdax, where A is a real skew-symmetric matrix or a real tridiagonal symmetric matrix with a constant diagonal. The algorithm uses only orthogonal similarity transformations and is believed to be the most efficient procedure available for computing all the eigenvalues or the complete eigensystem for the indicated classes of matrices. The three subroutines of the algorithm and their functions are described as follows: TRIZD--a subroutine that transforms an arbitrary real skew-symmetric matrix to skew-symmetric tridiagonal form by using orthogonal similarity transformations; IMZD--a subroutine that computes the eigenvalues and, optionally, the eigenvectors of a symmetric tridiagonal matrix with zeros on the diagonal or of a skew-symmetric tridiagonal matrix; TBAKZD--a subroutine that computes the eigenvectors of an arbitrary real skew-symmetric matrix by back-transforming the eigenvectors of the corresponding skew-symmetric tridiagonal matrix determined by TRIZD. The subroutines TRIZD, IMZD, and TBAKZD have been tested extensively on an IBM 360/91 computer using double precision arithmetic. Complete subroutine listings are available. (RWR)

On the convergence of the ordered roots of a sequence of determinantal equations

Given three sets of $n$ real eigenvalues satisfying the trace equality and the Horn inequalities, we know that there are $n\times n$ real symmetric matrices $A$ and $B$ so that $A$ has the first set of eigenvalues, $B$ has the second set of eigenvalues, and $A+B$ has the last set of eigenvalues. Under the condition that $B$ is a rank 2 matrix, we give a construction for the matrices $A$ and $B$. This construction is based on performing two orthogonal rank 1 updates on $A$. We end with a discussion of the relationship between this rank 2 Horn problem and the following similar problem: given a set of $n$ real eigenvalues, a set of $2$ real eigenvalues, and a set of $n+2$ real eigenvalues satisfying certain conditions, find an $(n+2)\times(n+2)$ real symmetric matrix such that the top left principal submatrix has the first set of eigenvalues, the bottom right principal submatrix has the second set of eigenvalues, and the full matrix has the last set of eigenvalues.

The inertia-preservers of several sets of matrices are identified. The sets include: all real matrices, all complex matrices, triangular matrices, real symmetric matrices and Hermitian matrices.

Many natural phenomena can be modeled by a second-order dynamical system , where stands for an appropriate state variable and M, C, K are time-invariant, real and symmetric matrices. In contrast to the classical inverse vibration problem where a model is to be determined from natural frequencies corresponding to various boundary conditions, the inverse mode problem concerns the reconstruction of the coefficient matrices (M, C, K) from a prescribed or observed subset of natural modes. This paper set forth a mathematical framework for the inverse mode problem and resolves some open questions raised in the literature. In particular, it shows that given merely the desirable structure of the spectrum, namely given the cardinalities of real or complex eigenvalues but not of the actual eigenvalues, the set of eigenvectors can be completed via solving an under-determined nonlinear system of equations. This completion suffices to construct symmetric coefficient matrices (M, C, K) whereas the underlying system can have arbitrary eigenvalues. Generic conditions under which the real symmetric quadratic inverse mode problem is solvable are discussed. Applications to important tasks such as updating models without spill-over or constructing models with positive semi-definite coefficient matrices are discussed.

THE PURPOSE of this paper is to develop relations among factors of raw scores, deviation scores, and double-centered scores. The treat ment is abstract and is merely an application of the algebra to the specialized matrices in which the interest of the factor analyst centers. Equiv alence of matrices, either square or rectangular, and similarity of square matrices are general notions that underlie the discussion. In addition, use is made of a specialized case of equivalence of matrices that occurs if the symmetric matrices generated by these matrices are elements of the same multiplicative group. Finally, a loose relation among matrices that generate symmetric matrices of different ranks belonging to the same sub-ring, or sub-algebra, of the total matric ring is employed to describe some of the results. It will be assumed that only real score matrices are employed and consequently that the symmetric ma trices they generate are not only real but also Gramian, that is, have no negative principal min ors. This is merely an arbitrary choice of fields, made because scores ordinarily are taken to be real numbers. The plan of the paper is first to describe a no tation that permits writing deviation scores as the matric product of raw scores and an idempo tent matrix. This notation should have a number of uses, a few of which are illustrated in this pa per. Next, the resolution of conventional devia tion scores into the product of factors and factor scores is summarized. This principle is, of course, well known; as it is summarized here, explicit use is made of the notion of groups o f singular matrices for which a symmetric idem potent matrix is a unit for multiplication. With the method established, it is possible to write equations relating the factors of raw scores to the factors of deviation scores, and equations re lating the factors of the two types of deviation scores. These results give a precise statement of Burt's reciprocity principle. Finally, double centered matrices are considered. Apparently two types of double-centered matrices may be identified; the relation of factors of such matrices to the factors of raw or deviation scores is out lined. It should be noted that an important restriction is made in this study. Throughout, it is assumed that the total variation exhibited in a set of data is to be analyzed in terms of common factors. This is a choice that the factor analyst may make. The results given here outline possibilities for relating solutions if he makes such a choice.

We consider the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a real symmetric random matrix. Our main result is that the existing result for a random matrix from the Gaussian Orthogonal Ensemble, obtained by Brézin and Hikami (2001), essentially continues to hold for a general real symmetric Wigner matrix. To obtain this result, we adapt the approach by Götze and Kösters (2008), who proved the analogous result for the Hermitian case.