Principal Submatrix Research Articles

The elliptic matrix completion problem

Completions of partial elliptic matrices are studied. Given an undirected graph G , it is shown that every partial elliptic matrix with graph G can be completed to an elliptic matrix if and only if the maximal cliques of G are pairwise disjoint. Further, given a partial elliptic matrix A with undirected graph G, it is proved that if G is chordal and each specified principal submatrix defined by a pair of intersecting maximal cliques is nonsingular, then A can be completed to an elliptic matrix. Conversely, if G is nonchordal or if the regularity condition is relaxed, it is shown that there exist partial elliptic matrices which are not completable to an elliptic matrix. In the process we obtain several results concerning chordal graphs that may be of independent interest.

Algebraic Multigrid for Linear Systems Obtained by Explicit Element Reduction

We consider sparse linear systems, where a set of “interior” degrees of freedom have been eliminated in order to reduce the problem size. This elimination is assumed to be local, so the “interior” principal submatrix is block-diagonal, and the resulting Schur complement is still sparse. For it to be beneficial, the elimination process should lead to reduced memory requirements, but equally importantly, it should also result in an algebraic problem that can be solved efficiently. In this paper we propose a general element reduction approach and show how the elimination process can exploit a particular “subzonal” discretization to maintain the sparsity of the Schur complement. We also investigate algebraic multigrid (AMG) solution algorithms applied to the reduced problem, and we discuss the influence of the local elimination on solver-related properties of the matrix, such as near-nullspace preservation and the availability of stable subspace decompositions. We focus on BoomerAMG, a parallel variant of classical Ruge–Stüben AMG, applied to scalar diffusion problems [V. Henson and U. Yang, Appl. Numer. Math., 41 (2002), pp. 155–177], and the auxiliary-space Maxwell solver (AMS) for electromagnetic diffusion applications [T. Kolev and P. Vassilevski, J. Comput. Math., 27 (2009), pp. 604–623]. In the electromagnetic case, we establish algebraically a reduced version of the Hiptmair–Xu decomposition from [R. Hiptmair and J. Xu, SIAM J. Numer. Anal., 45 (2007), pp. 2483–2509] and consider a modification of the reduction process that targets the singular problems arising in simulations with pure void (zero conductivity) regions. For scalar diffusion problems, our particular stencil analysis shows that the reduction has a positive effect on meshes with stretched elements. We present a number of two-dimensional, three-dimensional, and axisymmetric numerical experiments, which demonstrate that the combination of an appropriately chosen local elimination with the use of the BoomerAMG and AMS solvers can lead to significant improvements in the overall solution time.

Extreme spectra realization by real symmetric tridiagonal and real symmetric arrow matrices

We consider the following two problems: to construct a real symmetric arrow matrix A and to construct a real symmetric tridiagonal matrix A, from a special kind of spectral information: one eigenvalue � (j) of the j×j leading principal submatrix Aj of A, j = 1,2,...,n; and one eigenpair (� (n) ,x) of A. Here we give a solution to the first problem, introduced in (J. Peng, X.Y. Hu, and L. Zhang. Two inverse eigenvalue problems for a special kind of matrices. Linear Algebra Appl., 416:336-347, 2006.). In particular, for both problems to have a solution, we give a necessary and sufficient condition in the first case, and a sufficient conditio n in the second one. In both cases, we also give sufficient conditions in order that the constructedmatrices be nonnegative. Our results are constructive and they generate algorithmic procedures to construct such matrices.

Analogs of Cauchy–Poincaré and Fan–Pall interlacing theorems for [formula omitted]-Hermitian and [formula omitted]-normal matrices

The interlacing theorem of Cauchy–Poincaré states that the eigenvalues of a principal submatrix A 0 of a Hermitian matrix A interlace the eigenvalues of A . Fan and Pall obtained an analog of this theorem for normal matrices. In this note we investigate analogs of Cauchy–Poincaré and Fan–Pall interlacing theorems for J -Hermitian and J -normal matrices. The corresponding inverse spectral problems are also considered.

Non-singular acyclic matrices

Let A be an n by n real symmetric matrix, and let α be a non-empty subset of {1, …, n}. The principal submatrix of A obtained by removing rows and columns indexed by α is denoted by A(α). If the nullity of A(α) is |α| more than that of A, then α is called a P-set of A, and the indices in α are called P-vertices of A. This article concerns P-vertices and P-sets of non-singular acyclic matrices A when the graph T of A is a path or a star. It was shown in [C.R. Johnson and B.D. Sutton, Hermitian matrices, eigenvalue multiplicities, and eigenvector components, SIAM J. Matrix Anal. Appl. 26 (2004), pp. 390–399] that each singular acyclic matrix of order n has at most n − 2 P-vertices. In this article we show that this does not hold for non-singular acyclic matrices by constructing non-singular acyclic matrices, whose graphs are T, having n − 1 (or n) P-vertices (these matrices also achieve the maximum size of a P-set over non-singular acyclic matrices whose graphs are T). In addition, it is shown that if n is odd, then the upper bound n − 2 on the number of P-vertices of a singular acyclic matrix is tight, and (n − 1)/2 is the tight upper bound on the size of a P-set of a singular acyclic matrix.

A note on upper bounds on the cp-rank

Hanna and Laffey gave an upper bound on the cp-rank of a completely positive matrix, in terms of its rank and the number of zeros in a full rank principal submatrix. This bound, for the case that the matrix is positive, was improved by Barioli and Berman. In this paper a new straightforward proof of both results is given, and the same approach is used to improve Hanna and Laffey’s bound in the case that the matrix has a zero entry.

Formula omitted]-Analogs of distance matrices of 3-hypertrees

We consider the distance matrix of trees in 3-uniform hypergraphs (which we call 3-hypertrees). We give a formula for the inverse of a few q -analogs of distance matrices of 3-hypertrees T . Some results are analogs of results by Bapat et al. for graphs. We give an alternate proof of the result that the determinant of the distance matrix of a 3-hypertree T depends only on n , the number of vertices of T . Further, we give a Pfaffian identity for a principal submatrix of some (skew-symmetrized) distance matrices of 3-hypertrees when we fix an ordering of the vertices and assign signs appropriately. A result of Graham, Hoffman and Hosoya relates the determinant of the distance matrix of a graph and the determinants of its two-connected blocks. When the graph has as blocks a fixed connected graph H which satisfy some conditions, we give a formula for the inverse of its distance matrix. This result generalises a result of Graham and Lovasz. When each block of G is a fixed graph G , we also give some corollaries about the sum of the entries of the inverse of the distance matrix of G and some of its analogs.

On a recursive inverse eigenvalue problem

Let s 1, ..., s n be arbitrary complex scalars. It is required to construct an n × n normal matrix A such that s i is an eigenvalue of the leading principal submatrix A i , i = 1, 2, ..., n. It is shown that, along with the obvious diagonal solution diag(s 1, ..., s n ), this problem always admits a much more interesting nondiagonal solution A. As a rule, this solution is a dense matrix; with the diagonal solution, it shares the property that each submatrix A i is itself a normal matrix, which implies interesting connections between the spectra of the neighboring submatrices A i and A i + 1.

Gaussian elimination and the ranks of the components in the Cartesian decomposition of a matrix

Let A = B + iC, where B = B*, C = C*, be the Cartesian decomposition of an n × n matrix A, and let the component B (or C) have rank r < n. It is shown that for a nonsingular A, the inverse A−1 has an analogous property. This implies that all the (correctly defined) Schur complements in A have Cartesian decompositions with component B (or C) of rank ≤ r. The active submatrix at each step of the Gaussian elimination applied to A is the Schur complement of the appropriate leading principal submatrix. Bibliography: 2 titles.

Two inverse eigenproblems for symmetric doubly arrow matrices

In this paper, the problem of constructing a real symmetric doubly arrow matrix A from two special kinds of spectral information is considered. The first kind is the minimal and maximal eigenvalues of all leading principal submatrices of A, and the second kind is one eigenvalue of each leading principal submatrix of A together with one eigenpair of A. Sufficient conditions for both eigenproblems to have a solution and sufficient conditions for both eigenproblems have a nonnegative solution are given in this paper. The results are constructive in the sense that they generate algorithmic procedures to compute the solution matrix.

Inverse Eigenvalue Problem of Unitary Hessenberg Matrices

LetH∈ℂn×nbe ann×nunitary upper Hessenberg matrix whose subdiagonal elements are all positive, letHkbe thekthleading principal submatrix ofH, and letH˜kbe a modified submatrix ofHk. It is shown that when the minimal and maximal eigenvalues ofH˜k(k=1,2,…,n) are known,Hcan be constructed uniquely and efficiently. Theoretic analysis, numerical algorithm, and a small example are given.

The eigenvalue distribution on Schur complement of nonstrictly diagonally dominant matrices and general H-matrices

The paper studies the eigenvalue distribution of Schur complements of some specialmatrices, including nonstrictly diagonally dominant matrices and general H−matrices. Zhang, Xu, and Li [Theorem 4.1, The eigenvalue distribution on Schur complements of H-matrices. Linear Algebra Appl., 422:250–264, 2007] gave a condition for an n×n diagonally dominant matrix A to have |JR+(A)| eigenvalues with positive real part and |JR_ (A)| eigenvalues with negative real part, where |JR+ (A)| (|JR_ (A)|) denotes the number of diagonal entries of A with positive (negative) real part. This condition is applied to establish some results about the eigenvalue distribution for the Schur complements of nonstrictly diagonally dominant matrices and general H−matrices with complexdiagonal entries. Several conditions on the n × n matrix A and the subset α ⊆ N = {1, 2, · · · , n} arepresented so that the Schur complement A/α of A has |JR+ (A)|−|JαR+ (A)| eigenvalues with positivereal part and |JR_ (A)| − |JαR_ (A)| eigenvalues with negative real part, where |JαR+ (A)| (|JαR_ (A)|) denotes the number of diagonal entries of the principal submatrix A(α) of A with positive (negative) real part.

Branch duplication for the construction of multiple eigenvalues in an Hermitian matrix whose graph is a tree

Suppose that the eigenvalues of an Hermitian matrix A whose graph is a tree T are known, as well as the eigenvalues of the principal submatrix of A corresponding to a certain branch of T. A method for constructing a larger tree T ', in which the branch is ‘`duplicated’', and an Hermitian matrix A′ whose graph is T ' is described. The eigenvalues of A' are all of those of A, together with those corresponding to the branch, including multiplicities. This idea is applied (1) to give a solution to the inverse eigenvalue problem for stars, (2) to prove that the known diameter lower bound, for the minimum number of distinct eigenvalues among Hermitian matrices with a given graph, is best possible for trees of bounded diameter, and (3) to increase the list of trees for which all possible lists for the possible spectra are know. A generalization of the basic branch duplication method is presented.

Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products

For (n x n)-matrices A and B, define eta(A, B) = Sigma(S)det(A[S])det(B[S']), where the summation is over all subsets of {1,..., n}, S' is the complement of S', and A [S] is the principal submatrix ...

On Fiedler- and Parter-vertices of acyclic matrices

Let A be a real symmetric matrix and let λ be a real number. The algebraic multiplicity of λ as an eigenvalue of A is denoted by mA(λ), and the principal submatrix of A obtained by deleting row and column i from A is denoted by A(i). If mA(i)(λ)⩾mA(λ) (resp. mA(i)(λ)>mA(λ)), then index i is said to be a Fiedler-vertex (resp. a Parter-vertex) of A for λ. In this paper we provide geometric characterizations of Fiedler- and Parter-vertices of acyclic matrices, and give a geometric proof for the Parter–Wiener theorem in [C.R. Johnson, A. Leal Duarte, C.M. Saiago, The Parter–Wiener theorem: refinement and generalization, SIAM J. Matrix Anal. Appl. 25 (2003) 352–361]. Furthermore, we describe a structure of an acyclic matrix in terms of Fiedler- and Parter-vertices which enables us to construct an acyclic matrix of a desired form according to the locations of Fiedler- and Parter-vertices.

The inverse eigenvalue problem of reflexive matrices with a submatrix constraint and its approximation

In this paper, we first consider the existence of and the general expression for the solution to the constrained inverse eigenvalue problem defined as follows: given a generalized reflection matrix P∈Rn×n, a set of complex n-vectors {xi}i=1m, a set of complex numbers {λi}i=1m, and an s-by-s real matrix C0, find an n-by-n real reflexive matrix C such that the s-by-s leading principal submatrix of C is C0, and {xi}i=1m and {λi}i=1m are the eigenvectors and eigenvalues of C, respectively. We are then concerned with the best approximation problem for the constrained inverse problem whose solution set is nonempty. That is, given an arbitrary real n-by-n matrix \(\tilde{C}\) , find a matrix C which is the solution to the constrained inverse problem such that the distance between C and \(\tilde{C}\) is minimized in the Frobenius norm. We give an explicit solution and a numerical algorithm to the best approximation problem. An illustrative experiment is also presented.

Least squares solutions to AX = B for bisymmetric matrices under a central principal submatrix constraint and the optimal approximation

A matrix A ∈ R n × n is called a bisymmetric matrix if its elements a i , j satisfy the properties a i , j = a j , i and a i , j = a n - j + 1 , n - i + 1 for 1 ⩽ i , j ⩽ n . This paper considers least squares solutions to the matrix equation AX = B for A under a central principal submatrix constraint and the optimal approximation. A central principal submatrix is a submatrix obtained by deleting the same number of rows and columns in edges of a given matrix. We first discuss the specified structure of bisymmetric matrices and their central principal submatrices. Then we give some necessary and sufficient conditions for the solvability of the least squares problem, and derive the general representation of the solutions. Moreover, we also obtain the expression of the solution to the corresponding optimal approximation problem.

Spectra of principal submatrices of nonnegative matrices

Let A be an n × n nonnegative matrix with the spectrum ( λ 1 , λ 2 , … , λ n ) and let A 1 be an m × m principal submatrix of A with the spectrum ( μ 1 , μ 2 , … , μ m ) . In this paper we present some cases where the realizability of ( μ 1 , μ 2 , … , μ m , ν 1 , ν 2 , … , ν s ) implies the realizability of ( λ 1 , λ 2 , … , λ n , ν 1 , ν 2 , … , ν s ) and consider the question whether this holds in general. In particular, we show that the list ( λ 1 , λ 2 , … , λ n , - μ 1 , - μ 2 , … , - μ m ) is realizable.

The lower and upper bounds on Perron root of nonnegative irreducible matrices

Let A be an n × n nonnegative irreducible matrix, let A [ α ] be the principal submatrix of A based on the nonempty ordered subset α of { 1 , 2 , … , n } , and define the generalized Perron complement of A [ α ] by P t ( A / A [ α ] ) , i.e., P t ( A / A [ α ] ) = A [ β ] + A [ β , α ] ( tI - A [ α ] ) - 1 A [ α , β ] , t > ρ ( A [ α ] ) . This paper gives the upper and lower bounds on the Perron root of A. An upper bound on Perron root is derived from the maximum of the given parameter t 0 and the maximum of the row sums of P t 0 ( A / A [ α ] ) , synchronously, a lower bound on Perron root is expressed by the minimum of the given parameter t 0 and the minimum of the row sums of P t 0 ( A / A [ α ] ) . It is also shown how to choose the parameter t after α to get tighter upper and lower bounds of ρ ( A ) . Several numerical examples are presented to show that our method compared with the methods in [L.Z. Lu, M.K. Ng, Locations of Perron roots, Linear Algebra Appl. 392 (2004) 103–117.] is more effective.

An algorithm for computing Jordan chains and inverting analytic matrix functions

We present an efficient algorithm for obtaining a canonical system of Jordan chains for an n × n regular analytic matrix function A( λ) that is singular at the origin. For any analytic vector function b( λ), we show that each term in the Laurent expansion of A( λ) −1 b( λ) may be obtained from the previous terms by solving an ( n + d) × ( n+ d) linear system, where d is the order of the zero of det A( λ) at λ = 0. The matrix representing this linear system contains A(0) as a principal submatrix, which can be useful if A(0) is sparse. The last several iterations can be eliminated if left Jordan chains are computed in addition to right Jordan chains. The performance of the algorithm in floating point and exact (rational) arithmetic is reported for several test cases. The method is shown to be forward stable in floating point arithmetic.