Relative singular value decomposition and applications to LS-category

Abstract Let Sp ( n ) be the symplectic group of quaternionic ( n × n ) -matrices. For any 1 ≤ k ≤ n , an element A of Sp ( n ) can be decomposed in A = [ α T β P ] with P a ( k × k ) -matrix. In this work, starting from a singular value decomposition of P, we obtain what we call a relative singular value decomposition of A. This feature is well adapted for the study of the quaternionic Stiefel manifold X n , k , and we apply it to the determination of the Lusternik-Schnirelmann category of Sp ( k ) in X 2 k − j , k , for j = 0 , 1 , 2 .