The codimension of singular matrix pairs

Abstract The singular pairs of n × n matrices [those satisfying det( A − λB )  0] form a closed set of codimension n + 1 inside the space of all matrix pairs. The same holds for singular symmetric pairs. For Hermitian pairs, the singular ones form a closed set of codimension n + 1 or n + 2 according as n is odd or even. The irreducible components of these closed sets are determined by various basic singular summands.