The generalized null space decomposition (GNSD) is a unitary reduction of a general matrix A of order n to a block upper triangular form that reveals the structure of the Jordan blocks of A corresponding to a zero eigenvalue. The reduction was introduced by Kublanovskaya. It was extended first by Ruhe and then by Golub and Wilkinson, who based the reduction on the singular value decomposition. ...

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...

In this paper we construct a family of small unitary representations for real semisimple Lie groups associated with Jordan algebras. These representations are realized on L-spaces of certain orbits in the Jordan algebra. The representations are spherical and one of our key results is a precise L-estimate for the Fourier transform of the spherical vector. We also consider the tensor products of ...

It is an easy consequence of the Jordan canonical form that a matrix A ∈Mn×n(C) can be decomposed into a sum A = DA + NA where DA is a diagonalizable matrix, NA a nilpotent matrix, and such that DANA = NADA. It is clear that both DA and NA also commute with A. This decomposition is often referred to as the Jordan decomposition and has found many applications throughout the years. For example, i...