Toeplitz matrices with totally nonnegative inverses

It is shown that the inverse of a Toeplitz matrix has only nonnegative minors if the zeros of a certain polynomial are positive or if their arguments are less than π?(k+n), where n is the dimension and k+1 is the bandwidth of the matrix.     

On parametrization of totally nonpositive matrices and applications

The problem of accurate computations for totally non‐negative matrices has been studied; however, it remains open for other sign regular matrices. One major obstacle is that there is no known parametrization of these matrices. The main contribution of the present work is that we provide such parametrization of nonsingular totally nonpositive matrices. A useful application of our results is that these parameters can determine accurately the entries of the inverse of a nonsingular totally nonpositive matrix. Copyright © 2011 John Wiley & Sons, Ltd.     

A pivoting strategy for symmetric tridiagonal matrices

The LBL^T factorization of Bunch for solving linear systems involving a symmetric indefinite tridiagonal matrix T is a stable, efficient method. It computes a unit lower triangular matrix L and a block 1 × 1 and 2 × 2 matrix B such that T=LBL^T. Choosing the pivot size requires knowing a priori the largest element σ of T in magnitude. In some applications, it is required to factor T as it is formed without necessarily knowing σ. In this paper, we present a modification of the Bunch algorithm that can satisfy this requirement. We demonstrate that this modification exhibits the same bound on the growth factor as the Bunch algorithm and is likewise normwise backward stable. Copyright © 2005 John Wiley & Sons, Ltd.     

A simplified pivoting strategy for symmetric tridiagonal matrices

The pivoting strategy of Bunch and Marcia for solving systems involving symmetric indefinite tridiagonal matrices uses two different methods for solving 2 × 2 systems when a 2 × 2 pivot is chosen. In this paper, we eliminate this need for two methods by adding another criterion for choosing a 1 × 1 pivot. We demonstrate that all the results from the Bunch and Marcia pivoting strategy still hold. Copyright © 2006 John Wiley & Sons, Ltd.     

Norm estimates for the inverses of matrices of monotone type and totally positive matrices

We give estimates of the infinity norm of the inverses of matrices of monotone type and totally positive matrices.     

Singular values and diagonal elements of complex symmetric matrices

Necessary and sufficient conditions are given for the existence of a complex symmetric matrix with prescribed diagonal elements and singular values.     

On completions of symmetric and antisymmetric block diagonal partial matrices

A partial matrix is a matrix where only some of the entries are given. We determine the maximum rank of the symmetric completions of a symmetric partial matrix where only the diagonal blocks are given and the minimum rank and the maximum rank of the antisymmetric completions of an antisymmetric partial matrix where only the diagonal blocks are given.     

Factoring matrices with a tree-structured sparsity pattern

Let A be a matrix whose sparsity pattern is a tree with maximal degree d_max. We show that if the columns of A are ordered using minimum degree on |A|+|A|^∗, then factoring A using a sparse LU with partial pivoting algorithm generates only O(d_maxn) fill, requires only O(d_maxn) operations, and is much more stable than LU with partial pivoting on a general matrix. We also propose an even more efficient and just-as-stable algorithm called sibling-dominant pivoting. This algorithm is a strict partial pivoting algorithm that modifies the column preordering locally to minimize fill and work. It leads to only O(n) work and fill. More conventional column pre-ordering methods that are based (usually implicitly) on the sparsity pattern of |A|^∗|A| are not as efficient as the approaches that we propose in this paper.     

Band matrices with Toeplitz inverses

It is shown that a square band matrix H=(h_ij) with h_ij=0 for j? i>r and i?j>s, where r+s is less than the order of the matrix, has a Toeplitz inverse if and only if it has a special structure characterized by two polynomials of degrees r and s, respectively.     

Toeplitz matrices with banded inverses

Given a Toeplitz matrix T with banded inverse [i.e., (T^?1)_ij=0 for j?i>p], we show that the elements of T can be expressed in terms of the roots of a polynomial. Then, using properties we have previously established, we generalize this result appropriately to allow singular T and show that the converse also holds. Finally, we give a sufficient condition for the decay of the elements of T as one moves away from the diagonal.     

A theorem concerning certain sign symmetric matrices whose inverses are Morishima

Suppose A is an invertible sign symmetric matrix whose associated digraph D(A) is a tree. Then A^-1 will be Morishima iff a^?? ? 0 for all interior points ? in D(A). A^-1 will be anti-Morishima iff a^?? ? 0 for all interior points ? in D(A).     

Full rank factorization in echelon form of totally nonpositive (negative) rectangular matrices

An n×m real matrix A is said to be totally nonpositive (negative) if every minor is nonpositive (negative). In this paper, we study the full rank factorization in echelon form of a totally nonpositive (negative) matrix. This characterization allows us to significantly reduce the number of minors to be checked in order to decide the total negativity of a matrix.     

Toeplitz matrices with Toeplitz inverses revisited

A necessary and sufficient condition derived by Huang and Cline for a nonsingular Toeplitz matrix to have a Toeplitz inverse is shown to hold under more general hypotheses than indicated by them.     

Treediagonal matrices and their inverses

A generalization of tridiagonal matrices is considered, namely treediagonal matrices, which have nonzero off-diagonal elements only in positions where the adjacency matrix of a tree has nonzero elements. Some properties of treediagonal matrices are given, and their inverses are characterized and shown to have an interesting structure.     

Left inverses of matrices with polynomial decay

It is known that the algebra of Schur operators on ?² (namely operators bounded on both ?¹ and ?^∞) is not inverse-closed. When ?²=?²(X) where X is a metric space, one can consider elements of the Schur algebra with certain decay at infinity. For instance if X has the doubling property, then Q. Sun has proved that the weighted Schur algebra Aω(X) for a strictly polynomial weight ω is inverse-closed. In this paper, we prove a sharp result on left-invertibility of the these operators. Namely, if an operator A∈Aω(X) satisfies ‖Afp_‖?‖fp_‖, for some 1?p?∞, then it admits a left-inverse in Aω(X). The main difficulty here is to obtain the above inequality in ?². The author was both motivated and inspired by a previous work of Aldroubi, Baskarov and Krishtal (2008) [1], where similar results were obtained through different methods for X=Zd, under additional conditions on the decay.