We study Kostant’s partial order on the elements of a semisimple Lie group in relations with the finite dimensional representations. In particular, we prove the converse statement of [3, Theorem 6.1] on hyperbolic elements. A matrix in GLn(C) is called elliptic (resp. hyperbolic) if it is diagonalizable with norm 1 (resp. real positive) eigenvalues. It is called unipotent if all its eigenvalues...

Abstract We consider the possible Jordan canonical forms of Riordan arrays. prove that there are, in fact, only two such forms. Moreover, transition matrix is group case when given array has one some three specific

It is shown that an innnite Hankel matrix of a nite rank (or a nite Hankel matrix) admits a generalized Vandermonde decomposition H = V T DV , where V is a generalized Vandermonde matrix, and D is a block diagonal matrix. The full structure of this decomposition was rst fully discussed by Vandevoorde 9], but the development here is based solely on linear algebra considerations, speciically the ...

In this paper the problem of designing a single actuator control for a class of systems is addressed. The existence of such control is studied and several ways for designing such control are provided. The results depend either on the rational canonical form or on the Jordan structure associated with the matrix which characterizes the system dynamics. The constructed control laws can be employed...

Abstract. If an n× n complex matrix A is nonsingular, then for every integer m > 1, A has an mth root B, i.e., Bm = A. In this paper, we present a new simple proof for the Jordan canonical form of the root B. Moreover, a necessary and sufficient condition for the existence of mth roots of a singular complex matrix A is obtained. This condition is in terms of the dimensions of the null spaces of...

The eigenvectors and eigenvalues of block circulant matrices had been found for real symmetric matrices with symmetric submatrices, and for block circulant matrices with circulant submatrices. The eigenvectors are now found for general block circulant matrices, including the Jordan Canonical Form for defective eigenvectors. That analysis is applied to Stephen J. Watson’s alternating circulant m...

We study regular non-semisimple Dubrovin-Frobenius manifolds in dimensions 2,3,4. focus on the case where Jordan canonical form of operator multiplication by Euler vector field has a single block. Our results rely existence special local coordinates introduced [4] for flat F-manifolds with field. In such invariant metric manifold takes which is starting point our construction.

We consider the web hyperlink matrix used by Google for computing the PageRank whose form is given by A(c) = [cP + (1 − c)E]T , where P is a row stochastic matrix, E is a row stochastic rank one matrix, and c ∈ [0, 1]. We determine the analytic expression of the Jordan form of A(c) and, in particular, a rational formula for the PageRank in terms of c. The use of extrapolation procedures is very...

Relaxed Jordan canonical form (RJCF) is introduced for three-dimensional linear transformations, and applied to various problems in computer animation and visualization, especially problems involving matrix exponentials and matrix logarithms. In all cases our procedures use closed-form expressions, so they are not iterative, whereas the corresponding procedures that have been published in the g...