An involutory upper triangular Pascal matrix Un is investigated. Eigenvectors of Un and of UT n are considered. A characterization of Un is obtained, and it is shown how the results can be extended to matrices over a commutative ring with unity. © 2004 Elsevier Inc. All rights reserved.

We consider calculation of the dimensions of self-affine fractals and multifractals that are the attractors of iterated function systems specified in terms of upper triangular matrices. Using methods from linear algebra we obtain explicit formulae for the dimensions that are valid in many cases.

The structure of the group of quasi multiplicative arithmetical functions such that f(1) 6=0 with respect to Dirichlet and the more general Davison convolution via an isomorphism to a subgroup of upper triangular and Toeplitz matrices will be described. AMS Classification Number: 11A25

Let S be finite nonempty set of inequivlent valuations on Fp(t), and OS be the ring of S-integers. If Bn is the solvable, linear algebraic group of upper triangular matrices with determinant 1, then the solvable S-arithmetic group Bn(OS) has a finite index subgroup with infinite dimensional cohomology group in dimension |S|.

The invariants of solvable Lie algebras with nilradicals isomorphic to the algebra of strongly upper triangular matrices and diagonal nilindependent elements are studied exhaustively. Bases of the invariant sets of all such algebras are constructed by an original purely algebraic algorithm based on Cartan’s method of moving frames.

We prove a lower bound of 52n 2 3n for the rank of n n–matrix multiplication over an arbitrary field. Similar bounds hold for the rank of the multiplication in noncommutative division algebras and for the multiplication of upper triangular matrices.

Let T be an operator-weighted shift whose weights are 2-by-2 matrices. We say that, given > 0, T is in the -canonical form if each weight is an upper triangular matrix (aij), with 0 ≤ a11, a22 ≤ 1 and a12 6= 0 implies a11, a22 < . We generalize this concept to operator-weighted shifts whose weights are n-by-n matrices and we show that every polynomially bounded weighted shift, whose weights are...