CONTINUOUS BORDERING OF MATRICES AND CONTINUOUS MATRIX DECOMPOSITIONS

Let M(n) be the set of all real n x n-matrices of rank greater than or equal to n - 1. We prove that for n greater than or equal to 2 there are no continuous vector fields l,r : M(n) --> R(n) such that the bordered matrix [GRAPHICS] is regular for all A is an element of M(n). This result has some relevance for the numerical analysis of steady state bifurcation. As a by-product we show that there is no nonvanishing continuous vector field u : M(n)((n-1)) --> R(n) with Au(A) = 0 for all A is an element of M(n)((n-1)), where M(n)((n-1)) subset of M(n) is the set of all matrices of rank deficiency one. This implies that there is no singular value decomposition of A depending continuously on A in any matrix set which contains M(n)((n-1)). As another application we prove that in general there is no global analytic singular value decomposition for analytic matrix valued functions of more than one real variable.