Permuted graph matrices and their applications

Abstract

A permuted graph matrix is a matrix V ∈ C(m+n)×m such that every row of the m × m identity matrix Im appears at least once as a row of V. Permuted graph matrices can be used in some contexts in place of orthogonal matrices, for instance when giving a basis for a subspace µ ⊆ cm+nor to normalize matrix pencils in a suitable sense. In these applications the permuted graph matrix can be chosen with bounded entries, which is useful for stability reasons; several algorithms can be formulated with numerical advantage with permuted graph matrices.We present the basic theory and review some applications from optimization or in control theory.