; a ∈ A,m ∈ M, b ∈ B} equipped with the usual 2× 2 matrix-like addition and matrix-like multiplication is an algebra. An algebra T is called a triangular algebra if there exist algebras A and B and nonzero A−B-bimodule M such that T is (algebraically) isomorphic to Tri(A,M,B) under matrixlike addition and matrix-like multiplication; cf. [1]. For example, the algebra Tn of n × n upper triangular...

This research introduces a row compression and nested product decomposition of an n × n hierarchical representation of a rank structured matrix A, which extends the compression and nested product decomposition of a quasiseparable matrix. The hierarchical parameter extraction algorithm of a quasiseparable matrix is efficient, requiring only O(nlog(n)) operations, and is proven backward stable. T...

Abstract We describe the images of multilinear polynomials arbitrary degree evaluated on 3×3 upper triangular matrix algebra over an infinite field.

Given a real, symmetric matrix S, we define the slice FS through S as being the connected component containing S of two orbits under conjugation: the first by the orthogonal group, and the second by the upper triangular group. We describe some classical constructions in eigenvalue computations and integrable systems which keep slices invariant — their properties are clarified by the concept. We...

For little q-Jacobi polynomials, q-Hahn polynomials and big q-Jacobi polynomials we give particular q-hypergeometric series representations in which the termwise q = 0 limit can be taken. When rewritten in matrix form, these series representations can be viewed as decompositions into a lower triangular matrix times upper triangular matrix. We develop a general theory of such decompositions rela...