Some results concerning invertible Morishima and anti-Morishima matrices whose associated digraphs are trees

This paper addresses the following questions: Given that A is an invertible Morishima or anti-Morishima matrix, under what conditions will A^-1 be a Morishima or anti-Morishima matrix? Also, given the associated digraph of A and the fact that A, A^-1 are Morishima or anti-Morishima matrices, what can be said about the digraph of A^-1? A class of matrices is introduced where there questions are investigated.     

Some structural results concerning certain classes of qualitative matrices and their inverses

This paper presents some precise structural results concerning combinatorially symmetric, sign symmetric, and sign antisymmetric invertible matrices whose associated diagraphs are trees. In particular given an invertible sign antisymmetric matrix A whose associated digraph is a tree and the fact that A^-1 is sign antisymmetric, we are able to completely determine the associated digraph of A^-1.     

Group inverses of certain nonnegative matrices

Nonnegative matrices with the property that the group inverse of the matrixis equal to a power of the matrix are characterized. The special case of the Moore- Penrose inverse is considered. The situation when the matrix is stochastic is examined.     

Generalized inverses of certain Toeplitz matrices

The spectral inverse A^s of a Toeplitz matrix A whose form is related to that of a circulant matrix is studied by describing the algebraic structure of the semigroup of all matrices commuting with a given matrix with distinct eigenvalues. A computational form for A^s is given and necessary and sufficient conditions are found for A^s to be the Moore-Penrose generalized inverse A⁺.     

Exact inverses of certain band matrices

Summary A method for determining exact inverses for arbitrary size band matrices of Toeplitz type and closely related types is outlined. A number of examples arising in statistical problems and finite differences are considered and the initial required elements of the inverses are given.     

A min-max theorem for complex symmetric matrices

We optimize the form Re x^tTx to obtain the singular values of a complex symmetric matrix T. We prove that for ,

     

Block LU factorization is stable for block matrices whose inverses are block diagonally dominant

Let A M_n (C) and let the inverse matrix B = A^–1 be block diagonally dominant by rows (columns) w.r.t. an m × m block partitioning and a matrix norm. We show that A possesses a block LU factorization w.r.t. the same block partitioning, and the growth factor for A in this factorization is bounded above by 1 + , where = max _{1im i} and _i, 0 _i 1, are the row (column) block dominance factors of B. Further, the off-diagonal blocks of A (and of its block Schur complements) satisfy the inequalities

Inversion of certain symmetric band matrices

A procedure is given for inversion of a symmetric band matrix with all elements in a certain diagonal equal. Starting from the main diagonal we have the elementsk,k–1,k–2, ... 2,1 with zeros in the remaining diagonals. Letting the second order difference operator ² operate on the rows of the matrix we obtain a new matrix which can be inverted by a partition method after certain permutations of the elements.     

A general theorem concerning semi-groups with certain finiteness conditions

For an arbitrary abstract theoretico-semi-group propertyθ, satisfying certain natural conditions, we describe (to within the structure of groups possessing the propertyθ) the structure of periodic subgroups with the propertyθ.     

Generalized Hadamard matrices whose transposes are not generalized Hadamard matrices

In answer to “Research Problem 16” in Horadam's recent book Hadamard matrices and their applications, we provide a construction for generalized Hadamard matrices whose transposes are not generalized Hadamard matrices. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 456–458, 2009     

Doubly stochastic matrices whose squares are idempotent

Formulas for the doubly stochastic square roots of the idempotents are presented. Such square roots must necessarily satisfy per A²≤perA.     

Cochran's statistical theorem for outer inverses of matrices and matrix quadratic forms

We extend the matrix version of Cochran's statistical theorem to outer inverses of a matrix. As applications, we investigate the Wishartness and independence of matrix quadratic forms for Kronecker product covariance structures.     

Cochran's statistical theorem for outer inverses of matrices and matrix quadratic forms

We extend the matrix version of Cochran's statistical theorem to outer inverses of a matrix. As applications, we investigate the Wishartness and independence of matrix quadratic forms for Kronecker product covariance structures.     

Factoring symmetric totally nonpositive matrices and inverses with a diagonal pivoting method

In this paper, we consider how to factor symmetric totally nonpositive matrices and their inverses by taking advantage of the symmetric property. It is well-known that the Bunch-Kaufman algorithm is the most commonly used pivoting strategy which can, however, produce arbitrarily large entries in the lower triangular factor for such matrices as illustrated by our example. Therefore, it is interesting to show that when the Bunch-Parlett algorithm is simplified for these matrices, it only requires O(n ²) comparisons with the growth factor being nicely bounded by 4. These facts, together with a nicely bounded lower triangular factor and a pleasantly small relative backward error, show that the Bunch-Parlett algorithm is more preferable than the Bunch-Kaufman algorithm when dealing with these matrices.     

Allow problems concerning spectral properties of sign pattern matrices: A survey

An n×n sign pattern matrix has entries in {+,-,0}. This paper surveys the following problems concerning spectral properties of sign pattern matrices: sign patterns that allow all possible spectra (spectrally arbitrary sign patterns); sign patterns that allow all inertias (inertially arbitrary sign patterns); sign patterns that allow nilpotency (potentially nilpotent sign patterns); and sign patterns that allow stability (potentially stable sign patterns). Relationships between these four classes of sign patterns are given, and several open problems are identified.