Construction of Matrices with Prescribed Singular Values and Eigenvalues

Two issues concerning the construction of square matrices with prescribe singular values an eigenvalues are addressed. First, a necessary and sufficient condition for the existence of an n × n complex matrix with n given nonnegative numbers as singular values an m ( n) given complex numbers to be m of the eigenvalues is determined. This extends the classical result of Weyl and Horn treating the case when m = n. Second, an algorithm is given to generate a triangular matrix with prescribe singular values an eigenvalues. Unlike earlier algorithms, the eigenvalues can be arranged in any prescribe order on the diagonal. A slight modification of this algorithm allows one to construct a real matrix with specified real an complex conjugate eigenvalues an specified singular values. The construction is done by multiplication by diagonal unitary matrices, permutation matrices and rotation matrices. It is numerically stable and may be useful in developing test software for numerical linear algebra packages.     

Numerical enclosure for multiple eigenvalues of an Hermitian matrix whose graph is a tree

In this paper we consider a numerical enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. If an Hermitian matrix A whose graph is a tree has multiple eigenvalues, it has the property that matrices which are associated with some branches in the undirected graph of A have the same eigenvalues. By using this property and interlacing inequalities for Hermitian matrices, we show an enclosure method for multiple eigenvalues of an Hermitian matrix whose graph is a tree. Since we do not generally know whether a given matrix has exactly a multiple eigenvalue from approximate computations, we use the property of interlacing inequalities to enclose some eigenvalues including multiplicities.In this process, we only use the enclosure of simple eigenvalues to enclose a multiple eigenvalue by using a computer and interval arithmetic.     

Computing eigenspaces with specified eigenvalues of a regular matrix pair (A,B) and condition estimation: theory,algorithms and software

Theory, algorithms and LAPACK-style software for computing a pair of deflating subspaces with specified eigenvalues of a regular matrix pair (A, B) and error bounds for computed quantities (eigenvalues and eigenspaces) are presented. Thereordering of specified eigenvalues is performed with a direct orthogonal transformation method with guaranteed numerical stability. Each swap of two adjacent diagonal blocks in the real generalized Schur form, where at least one of them corresponds to a complex conjugate pair of eigenvalues, involves solving a generalized Sylvester equation and the construction of two orthogonal transformation matrices from certain eigenspaces associated with the diagonal blocks. The swapping of two 1×1 blocks is performed using orthogonal (unitary) Givens rotations. Theerror bounds are based on estimates of condition numbers for eigenvalues and eigenspaces. The software computes reciprocal values of a condition number for an individual eigenvalue (or a cluster of eigenvalues), a condition number for an eigenvector (or eigenspace), and spectral projectors onto a selected cluster. By computing reciprocal values we avoid overflow. Changes in eigenvectors and eigenspaces are measured by their change in angle. The condition numbers yield bothasymptotic andglobal error bounds. The asymptotic bounds are only accurate for small perturbations (E, F) of (A, B), while the global bounds work for all (E, F.) up to a certain bound, whose size is determined by the conditioning of the problem. It is also shown how these upper bounds can be estimated. Fortran 77software that implements our algorithms for reordering eigenvalues, computing (left and right) deflating subspaces with specified eigenvalues and condition number estimation are presented. Computational experiments that illustrate the accuracy, efficiency and reliability of our software are also described.     

Permutation Matrices, Wreath Products, and the Distribution of Eigenvalues

We consider a class of random matrix ensembles which can be constructed from the random permutation matrices by replacing the nonzero entries of the n×n permutation matrix matrix with M×M diagonal matrices whose entries are random Kth roots of unity or random points on the unit circle. Let X be the number of eigenvalues lying in a specified arc I of the unit circle, and consider the standardized random variable (X–E[X])/(Var(X))^1/2. We show that for a fixed set of arcs I ₁,...,I _N , the corresponding standardized random variables are jointly normal in the large n limit, and compare the covariance structures which arise with results for other random matrix ensembles.     

The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law

LetAbe annbynmatrix whose elements are independent random variables with standard normal distributions. Girko's (more general) circular law states that the distribution of appropriately normalized eigenvalues is asymptotically uniform in the unit disk in the complex plane. We derive the exact expected empirical spectral distribution of the complex eigenvalues for finiten, from which convergence in the expected distribution to the circular law for normally distributed matrices may be derived. Similar methodology allows us to derive a joint distribution formula for the real Schur decomposition ofA. Integration of this distribution yields the probability thatAhas exactlykreal eigenvalues. For example, we show that the probability thatAhas all real eigenvalues is exactly 2^{−n(n−1)/4}.     

A property of eigenvalue bounds for a class of symmetric tridiagonal interval matrices

The eigenvalue bounds of interval matrices are often required in some mechanical and engineering fields. In this paper, we consider an interval eigenvalue problem with symmetric tridiagonal matrices. A theoretical result is obtained that under certain assumptions the upper and lower bounds of interval eigenvalues of the problem must be achieved just at some vertex matrices of the interval matrix. Then a sufficient condition is provided to guarantee the assumption to be satisfied. The conclusion is illustrated also by a numerical example. Copyright © 2010 John Wiley & Sons, Ltd.     

Scaled Optimal Path Trust-Region Algorithm

Trust-region algorithms solve a trust-region subproblem at each iteration. Among the methods solving the subproblem, the optimal path algorithm obtains the solution to the subproblem in full-dimensional space by using the eigenvalues and eigenvectors of the system. Although the idea is attractive, the existing optimal path method seems impractical because, in addition to factorization, it requires either the calculation of the full eigensystem of a matrix or repeated factorizations of matrices at each iteration. In this paper, we propose a scaled optimal path trust-region algorithm. The algorithm finds a solution of the subproblem in full-dimensional space by just one Bunch–Parlett factorization for symmetric matrices at each iteration and by using the resulting unit lower triangular factor to scale the variables in the problem. A scaled optimal path can then be formed easily. The algorithm has good convergence properties under commonly used conditions. Computational results for small-scale and large-scale optimization problems are presented which show that the algorithm is robust and effective.     

Pseudospectra of isospectrally reduced matrices

An isospectral matrix reduction is a procedure that reduces the size of a matrix while maintaining its eigenvalues up to a known set. As to not violate the fundamental theorem of algebra, the reduced matrices have rational functions as entries. Because isospectral reductions can preserve the spectrum of a matrix, they are fundamentally different from say the restriction of a matrix to an invariant subspace. We show that the notion of pseudospectrum can be extended to a wide class of matrices with rational function entries and that the pseudospectrum of such matrices shrinks with isospectral reductions. Hence, the eigenvalues of a reduced matrix are more robust to entry‐wise perturbations than the eigenvalues of the original matrix. Moreover, the isospectral reductions considered here are more general than those considered elsewhere. We also introduce the notion of an inverse pseudospectrum (or pseudoresonances), which indicates how stable the poles of a rational function valued matrix are to entry‐wise perturbations. Illustrations of these concepts are given for mass‐spring networks. Copyright © 2014 John Wiley & Sons, Ltd.     

Completion problem with partial correlation vines

This paper extends the results in [D. Kurowicka, R.M. Cooke, A parametrization of positive definite matrices in terms of partial correlation vines, Linear Algebra Appl. 372 (2003) 225-251]. We show that a partial correlation vine represents a factorization of the determinant of the correlation matrix. We show that the graph of an incompletely specified correlation matrix is chordal if and only if it can be represented as an m-saturated incomplete vine, that is, an incomplete vine for which all edges corresponding to membership-descendents (m-descendents for short) of a specified edge are specified. This enables us to find the set of completions, and also the completion with maximal determinant for matrices corresponding to chordal graphs.     

Moore-Penrose Inverses and Group Inverses of Block k-Circulant Matrices

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The Sylvester–Kac matrix space

The Sylvester–Kac matrix is a tridiagonal matrix with integer entries and integer eigenvalues that appears in a variety of applicative problems. We show that it belongs to a four dimensional linear space of tridiagonal matrices that can be simultaneously reduced to triangular form. We name this space after the matrix.     

Computing a logarithm of a unitary matrix with general spectrum

We analyze an algorithm for computing a skew‐Hermitian logarithm of a unitary matrix and also skew‐Hermitian approximate logarithms for nearly unitary matrices. This algorithm is very easy to implement using standard software, and it works well even for unitary matrices with no spectral conditions assumed. Certain examples, with many eigenvalues near ? 1, lead to very non‐Hermitian output for other basic methods of calculating matrix logarithms. Altering the output of these algorithms to force skew‐Hermitian output creates accuracy issues, which are avoided by the considered algorithm. A modification is introduced to deal properly with the J‐skew‐symmetric unitary matrices. Applications to numerical studies of topological insulators in two symmetry classes are discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Asymptotic distribution of Wishart matrix for block-wise dispersion of population eigenvalues

This paper deals with the asymptotic distribution of Wishart matrix and its application to the estimation of the population matrix parameter when the population eigenvalues are block-wise infinitely dispersed. We show that the appropriately normalized eigenvectors and eigenvalues asymptotically generate two Wishart matrices and one normally distributed random matrix, which are mutually independent. For a family of orthogonally equivariant estimators, we calculate the asymptotic risks with respect to the entropy or the quadratic loss function and derive the asymptotically best estimator among the family. We numerically show (1) the convergence in both the distributions and the risks are quick enough for a practical use, (2) the asymptotically best estimator is robust against the deviation of the population eigenvalues from the block-wise infinite dispersion.     

Über Matrizen mit Semidefinitem Realteil, Imaginärteil oder Polarem Defekt und Ihre Eigenwerte

We give a new short proof using properties of the field of values to show that

a) a complex matrix with only real eigenvalues is either hermitian or has indefinite imaginary part, and

b) one with only purely imaginary eigenvalues is either skew-hermitian or has indefinite real part, while

c) one whose eigenvalues all have absolute value 1 is either unitary or has indefinite polar defect I—TT^*.

Conversely, every skewsymmetric matrix is the skewsymmetric part of some real matrix that is similar to a real diagonal matrix. The corresponding result for complex matrices is found to be false.     

Function of a square matrix with repeated eigenvalues

An analytical function f(A) of an arbitrary n×n constant matrix A is determined and expressed by the “fundamental formula”, the linear combination of constituent matrices. The constituent matrices Z_kh, which depend on A but not on the function f(s), are computed from the given matrix A, that may have repeated eigenvalues. The associated companion matrix C and Jordan matrix J are then expressed when all the eigenvalues with multiplicities are known. Several other related matrices, such as Vandermonde matrix V, modal matrix W, Krylov matrix K and their inverses, are also derived and depicted as in a 2-D or 3-D mapping diagram. The constituent matrices Z_kh of A are thus obtained by these matrices through similarity matrix transformations. Alternatively, efficient and direct approaches for Z_kh can be found by the linear combination of matrices, that may be further simplified by writing them in “super column matrix” forms. Finally, a typical example is provided to show the merit of several approaches for the constituent matrices of a given matrix A.     

On the eigenvalues of double band matrices

We consider matrices containing two diagonal bands of positive entries. We show that all eigenvalues of such matrices are of the form rζ, where r is a nonnegative real number and ζ is a pth root of unity, where p is the period of the matrix, which is computed from the distance between the bands. We also present a problem in the asymptotics of spectra in which such double band matrices are perturbed by banded matrices.     

Numerically stable deflation of hessenberg and symmetric tridiagonal matrices

While numerically stable techniques have been available for deflating a fulln byn matrix, no satisfactory finite technique has been known which preserves Hessenberg form. We describe a new algorithm which explicitly deflates a Hessenberg matrix in floating point arithmetic by means of a sequence of plane rotations. When applied to a symmetric tridiagonal matrix, the deflated matrix is again symmetric tridiagonal. Repeated deflation can be used to find an orthogonal set of eigenvectors associated with any selection of eigenvalues of a symmetric tridiagonal matrix.     

Random block matrices generalizing the classical Jacobi and Laguerre ensembles

In this paper we consider random block matrices which generalize the classical Laguerre ensemble and the Jacobi ensemble. We show that the random eigenvalues of the matrices can be uniformly approximated by the zeros of matrix orthogonal polynomials and obtain a rate for the maximum difference between the eigenvalues and the zeros. This relation between the random block matrices and matrix orthogonal polynomials allows a derivation of the asymptotic spectral distribution of the matrices.     

Eigenvalue condition numbers: Zero-structured versus traditional

We discuss questions of eigenvalue conditioning. We study in some depth relationships between the classical theory of conditioning and the theory of the zero-structured conditioning, and we derive from the existing theory formulae for the mathematical objects involved. Then an algorithm to compare the zero-structured individual condition numbers of a set of simple eigenvalues with the traditional ones is presented. Numerical tests are reported to highlight how the algorithm provides interesting information about eigenvalue sensitivity when the perturbations in the matrix have an arbitrarily assigned zero-structure. Patterned matrices (Toeplitz and Hankel) will be investigated in a forthcoming paper (Eigenvalue patterned condition numbers: Toeplitz and Hankel cases, Tech. Rep. 3, Mathematics Department, University of Rome ‘ La Sapienza’ , 2005.).     

On the eigenvalues of quaternion matrices

This article is a continuation of the article [F. Zhang, Ger?gorin type theorems for quaternionic matrices, Linear Algebra Appl. 424 (2007), pp. 139–153] on the study of the eigenvalues of quaternion matrices. Profound differences in the eigenvalue problems for complex and quaternion matrices are discussed. We show that Brauer's theorem for the inclusion of the eigenvalues of complex matrices cannot be extended to the right eigenvalues of quaternion matrices. We also provide necessary and sufficient conditions for a complex square matrix to have infinitely many left eigenvalues, and analyse the roots of the characteristic polynomials for 2?×?2 matrices. We establish a characterisation for the set of left eigenvalues to intersect or be part of the boundary of the quaternion balls of Ger?gorin.