Finding off‐diagonal entries of the inverse of a large symmetric sparse matrix

The method fast inverse using nested dissection (FIND) was proposed to calculate the diagonal entries of the inverse of a large sparse symmetric matrix. In this paper, we show how the FIND algorithm can be generalized to calculate off‐diagonal entries of the inverse that correspond to ‘short’ geometric distances within the computational mesh of the original matrix. The idea is to extend the downward pass in FIND that eliminates all nodes outside of each node cluster. In our advanced downwards pass, it eliminates all nodes outside of each ‘node cluster pair’ from a subset of all node cluster pairs. The complexity depends on how far (i,j) is from the main diagonal. In the extension of the algorithm, all entries of the inverse that correspond to vertex pairs that are geometrically closer than a predefined length limit l will be calculated. More precisely, let α be the total number of nodes in a two‐dimensional square mesh. We will show that our algorithm can compute O(α^{3 ∕ 2 + 2ε}) entries of the inverse in O(α^{3 ∕ 2 + 2ε}) time where l = O(α^{1 ∕ 4 + ε}) and 0 ≤ ε ≤1 ∕ 4. Numerical examples are given to illustrate the efficiency of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Positive definite completions of partial Hermitian matrices

The question of which partial Hermitian matrices (some entries specified, some free) may be completed to positive definite matrices is addressed. It is shown that if the diagonal entries are specified and principal minors, composed of specified entries, are positive, then, if the undirected graph of the specified entries is chordal, a positive definite completion necessarily exists. Furthermore, if this graph is not chordal, then examples exist without positive definite completions. In case a positive definite completion exists, there is a unique matrix, in the class of all positive definite completions, whose determinant is maximal, and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial Hermitian matrix. Additional observations regarding positive definite completions are made.     

Theorems on partitioned matrices revisited and their applications to graph spectra

Some old results about spectra of partitioned matrices due to Goddard and Schneider or Haynsworth are re-proved. A new result is given for the spectrum of a block-stochastic matrix with the property that each off-diagonal block has equal entries and each diagonal block has equal diagonal entries and equal off-diagonal entries. The result is applied to the study of the spectra of the usual graph matrices by partitioning the vertex set of the graph according to the neighborhood equivalence relation. The concept of a reduced graph matrix is introduced. The question of when n-2 is the second largest signless Laplacian eigenvalue of a connected graph of order n is treated. A recent conjecture posed by Tam, Fan and Zhou on graphs that maximize the signless Laplacian spectral radius over all (not necessarily connected) graphs with given numbers of vertices and edges is refuted. The Laplacian spectrum of a (degree) maximal graph is reconsidered.     

Matricial formulae for partitions

The exponential of the triangular matrix whose entries in the diagonal at distance n from the principal diagonal are all equal to the sum of the inverses of the divisors of n is the triangular matrix whose entries in the diagonal at distance n from the principal diagonal are all equal to the number of partitions of n. A similar result holds for all pairs of sequences satisfying a special mutual recurrence.     

Bounds for the Entries of Matrix Functions with Applications to Preconditioning

Let A be a symmetric matrix and let f be a smooth function defined on an interval containing the spectrum of A. Generalizing a well-known result of Demko, Moss and Smith on the decay of the inverse we show that when A is banded, the entries of f(A)are bounded in an exponentially decaying manner away from the main diagonal. Bounds obtained by representing the entries of f(A)in terms of Riemann-Stieltjes integrals and by approximating such integrals by Gaussian quadrature rules are also considered. Applications of these bounds to preconditioning are suggested and illustrated by a few numerical examples.     

The copositive completion problem: Unspecified diagonal entries

In [L. Hogben, C.R. Johnson, R. Reams, The copositive matrix completion problem, Linear Algebra Appl. 408 (2005) 207-211] it was shown that any partial (strictly) copositive matrix all of whose diagonal entries are specified can be completed to a (strictly) copositive matrix. In this note we show that every partial strictly copositive matrix (possibly with unspecified diagonal entries) can be completed to a strictly copositive matrix, but there is an example of a partial copositive matrix with an unspecified diagonal entry that cannot be completed to a copositive matrix.     

The Spectral Theory of Upper Triangular Matrices with Entries in a Banach Algebra

Assume that T is an upper triangular square matrix with entries in a unital Banach algebra. The main question studied here is: Under what conditions on the entries in T is it true that the spectrum of T is the union of the spectra of the diagonal entries of T? Also some results are proved concerning the Fredholm theroy of matrices with operator entries.     

Diagonal scaling in eigenvalue localization for the Schur complement

Geršgorin theorem is a well-known result in eigenvalue localization area. In this paper, using diagonal scaling method, we obtain more Geršgorin-type localizations for the eigenvalues of the Schur complement using the entries of the original matrix instead of the entries of the Schur complement. We deal with classes of matrices with some form of diagonal dominance. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Isospectral flow method for nonnegative inverse eigenvalue problem with prescribed structure

The nonnegative inverse eigenvalue problem is that given a family of complex numbers λ={λ₁,…,λ_n}, find a nonnegative matrix of order n with spectrum λ. This problem is difficult and remains unsolved partially. In this paper, we focus on its generalization that the reconstructed nonnegative matrices should have some prescribed entries. It is easy to see that this new problem will come back to the common nonnegative inverse eigenvalue problem if there is no constraint of the locations of entries. A numerical isospectral flow method which is developed by hybridizing the optimization theory and steepest descent method is used to study the reconstruction. Moreover, an error estimate of the numerical iteration for ordinary differential equations on the matrix manifold is presented. After that, a numerical method for the nonnegative symmetric inverse eigenvalue problem with prescribed entries and its error estimate are considered. Finally, the approaches are verified by the numerical test results.     

Some improvements for two-sided bounds on the inverse of diagonally dominant tridiagonal matrices

In this paper, we obtain lower and upper bounds for the entries of the inverses of diagonally dominant tridiagonal matrices. First of all we derive the bounds for off-diagonal elements of the inverse as a function of the diagonal ones, then we improve the two-sided bounds for the diagonal entries obtaining sharper lower and upper bounds for all the elements of the inverse.     

Positive matrix semigroups with binary diagonals

We show that a semigroup of positive matrices (all entries greater than or equal to zero) with binary diagonals (diagonal entries either 0 or 1) is either decomposable (all matrices in the semigroup have a common zero entry) or is similar, via a positive diagonal matrix, to a binary semigroup (all entries 0 or 1). In the case where the idempotents of minimal rank in S{\mathcal{S}} satisfy a “diagonal disjointness” condition, we obtain additional structural information. In the case where the semigroup is not necessarily positive but has binary diagonals we show that either the semigroup is reducible or the minimal rank ideal is a binary semigroup. We also give generalizations of these results to operators acting on the Hilbert space of square-summable sequences.     

Minimal paramount matrices

A real symmetric matrix of order n, n ? 2, is said to be paramount if each proper principal minor is not less than the absolute value of any other minor built from the same rows. A paramount matrix is minimal ¹ if reducing any of the diagonal entries removes the matrix from the paramount class. Minimal paramount matrices arise in the n-port realization problem of circuit theory. A condition is found that is equivalent to the minimality of a paramount matrix. Conditions are also found that guarantee that the inverse of an invertible minimal paramount matrix is itself minimal.     

Geršgorin variations III: On a theme of Brualdi and Varga

Brualdi brought to Geršgorin Theory the concept that the digraph G(A) of a matrix A is important in studying whether A is singular. He proved, for example, that if, for every directed cycle of G(A), the product of the diagonal entries exceeds the product of the row sums of the moduli of the off-diagonal entries, then the matrix is nonsingular. We will show how, in polynomial time, that condition can be tested and (if satisfied) produce a diagonal matrix D, with positive diagonal entries, such that AD (where A is any nonnnegative matrix satisfying the conditions) is strictly diagonally dominant (and so, A is nonsingular). The same D works for all matrices satisfying the conditions. Varga raised the question of whether Brualdi’s conditions are sharp. Improving Varga’s results, we show, if G is scwaltcy (strongly connected with at least two cycles), and if the Brualdi conditions do not hold, how to construct (again in polynomial time) a complex matrix whose moduli satisfy the given specifications, but is singular.     

On a conjugate directions method for solving strictly convex QP problem

Problem of solving the strictly convex, quadratic programming problem is studied. The idea of conjugate directions is used. First we assume that we know the set of directions conjugate with respect to the hessian of the goal function. We apply n simultaneous directional minimizations along these conjugate directions starting from the same point followed by the addition of the directional corrections. Theorem justifying that the algorithm finds the global minimum of the quadratic goal function is proved. The way of effective construction of the required set of conjugate directions is presented. We start with a vector with zero value entries except the first one. At each step new vector conjugate to the previously generated is constructed whose number of nonzero entries is larger by one than in its predecessor. Conjugate directions obtained by means of the above construction procedure with appropriately selected parameters form an upper triangular matrix which in exact computations is the Cholesky factor of the inverse of the hessian matrix. Computational cost of calculating the inverse factorization is comparable with the cost of the Cholesky factorization of the original second derivative matrix. Calculation of those vectors involves exclusively matrix/vector multiplication and finding an inverse of a diagonal matrix. Some preliminary computational results on some test problems are reported. In the test problems all symmetric, positive definite matrices with dimensions from \(14\times 14\) to \(2000\times 2000\) from the repository of the Florida University were used as the hessians.     

Vertex degrees and doubly stochastic graph matrices

In this article, the relationship between vertex degrees and entries of the doubly stochastic graph matrix has been investigated. In particular, we present an upper bound for the main diagonal entries of a doubly stochastic graph matrix and investigate the relations between a kind of distance for graph vertices and the vertex degrees. These results are used to answer in negative Merris' question on doubly stochastic graph matrices. These results may also be used to establish relations between graph structure and entries of doubly stochastic graph matrices. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:104‐114, 2011     

Noncirculant Toeplitz matrices all of whose powers are Toeplitz

Let a, b and c be fixed complex numbers. Let M _n (a, b, c) be the n × n Toeplitz matrix all of whose entries above the diagonal are a, all of whose entries below the diagonal are b, and all of whose entries on the diagonal are c. For 1 ⩽ k ⩽ n, each k × k principal minor of M _n (a, b, c) has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of M _n (a, b, c). We also show that all complex polynomials in M _n (a, b, c) are Toeplitz matrices. In particular, the inverse of M _n (a, b, c) is a Toeplitz matrix when it exists.     

Bounds on the inverses of nonnegative triangular matrices with monotonicity properties

Some bounds on the entries and on the norm of the inverse of triangular matrices with nonnegative and monotone entries are found. All the results are obtained by exploiting the properties of the fundamental matrix of the recurrence relation which generates the sequence of the entries of the inverse matrix. One of the results generalizes a theorem contained in a recent article of one of the authors about Toeplitz matrices.     

Bounds on the inverses of nonnegative triangular matrices with monotonicity properties

Some bounds on the entries and on the norm of the inverse of triangular matrices with nonnegative and monotone entries are found. All the results are obtained by exploiting the properties of the fundamental matrix of the recurrence relation which generates the sequence of the entries of the inverse matrix. One of the results generalizes a theorem contained in a recent article of one of the authors about Toeplitz matrices.     

A Galerkin boundary element method based on interpolatory Hermite trigonometric wavelets

A Galerkin boundary element method based on interpolatory Hermite trigonometric wavelets is presented for solving 2-D potential problems defined inside or outside of a circular boundary in this paper. In this approach, an equivalent variational form of the corresponding boundary integral equation for the potential problem is used; the trigonometric wavelets are employed as trial and test functions of the variational formulation. The analytical formulae of the matrix entries indicate that most of the matrix entries are naturally zero without any truncation technique and the system matrix is a block diagonal matrix. Each block consists of four circular submatrices. Hence the memory spaces and computational complexity of the system matrix are linear scale. This approach could be easily coupled into domain decomposition method based on variational formulation. Finally, the error estimates of the approximation solutions are given and some test examples are presented.     

Approximation of eigenvalues for unbounded Jacobi matrices using finite submatrices

We consider an infinite Jacobi matrix with off-diagonal entries dominated by the diagonal entries going to infinity. The corresponding self-adjoint operator J has discrete spectrum and our purpose is to present results on the approximation of eigenvalues of J by eigenvalues of its finite submatrices.