For matroids M and N on disjoint sets S and T , a semidirect sum of M and N is any matroid K on S ∪ T that, like the direct sum and the free product, has the restrictionK|S equal toM and the contractionK/S equal toN . We abstract a matrix construction to get a general matroid construction: the matroid union of any rank-preserving extension ofM on the set S ∪ T with the direct sum ofN and the ra...

Let p be a polynomial in non-commutative variables x1,x2,…,xn with constant term zero over an algebraically closed field K. The object of study this paper is the image kind algebra upper triangular matrices Tm(K). We introduce family polynomials called multi-index p-inductive for given p. Using we will show that, if identity Tt(K) but not Tt+1(K), then p(Tm(K))⊆Tm(K)(t−1). Equality achieved cas...

New perturbation analyses are presented for the block Cholesky downdating problem U T U = R T R ? X T X. These show how changes in R and X alter the Cholesky factor U. There are two main cases for the perturbation matrix R in R: (1) R is a general matrix; (2))R is an upper triangular matrix. For both cases, rst order perturbation bounds for the downdated Cholesky factor U are given using two ap...

The structure of graded triangular algebras T of arbitrary dimension are studied in this paper. This is motivated in part for the important role that triangular algebras play in the study of oriented graphs, upper triangular matrix algebras or nest algebras. It is shown that T decomposes as T = U + ( ∑ i∈I Ti), where U is an R-submodule contained in the 0-homogeneous component and any Ti a well...

We describe a new extension to ScaLAPACK [2] for computing with symmetric (Hermi-tian) matrices stored in a packed form. The new code is built upon the ScaLAPACK routines for full dense storage for a high degree of software reuse. The original ScaLAPACK stores a symmetric matrix as a full matrix but accesses only the lower or upper triangular part. The new code enables more efficient use of mem...

We investigate the affine canonical basis ([L2]) and the monomial basis constructed in [LXZ] in Lusztig’s geometric setting. We show that the transition matrix between the two bases is upper triangular with 1’s in the diagonal and coefficients in the upper diagonal entries in Z≥0[v, v ]. As a consequence, we show that part of the monomial basis elements give rise to resolutions of support varie...

We describe a new extension to ScaLAPACK [2] for computing with symmetric (Hermi-tian) matrices stored in a packed form. The new code is built upon the ScaLAPACK routines for full dense storage for a high degree of software reuse. The original ScaLAPACK stores a symmetric matrix as a full matrix but accesses only the lower or upper triangular part. The new code enables more efficient use of mem...

A tridiagonal matrix with entries given by square matrices is a block tridiagonal matrix; the matrix is banded if off-diagonal blocks are upper or lower triangular. Such matrices are of great importance in numerical analysis and physics, and to obtain general properties is of great utility. The blocks of the inverse matrix of a block tridiagonal matrix can be factored in terms of two sets of ma...

In this manuscript we reduce the computation of the singular values of a general product/quotient of matrices to the computation of the singular values of an upper triangular semiseparable matrix. Compared to the reduction into a bidiagonal matrix the reduction into semiseparable form exhibits a nested subspace iteration. Hence, when there are large gaps between the singular values, these gaps ...

This paper presents a menu-driven RATS-program which allows to identify structural shocks in vector-autoregressive (VAR) models. Identification is achieved by imposing short-run or long-run restrictions (or a combination of both) on the structural form of a model. The only requirement is that the matrix of restrictions consisting of rows of the restricted matrices of short-run and long-run effe...