On principal angles between subspaces in Rn

Abstract

Let L, M be subspaces in Rn, dim L = l≤dim M = m. Then the principal angles between L and M, 0≤θ1≤θ2≤⋯≤θ l≤π/2, are given by cosθi= 〈xiyi〉 ∥xi∥∥yi∥=max 〈x,y〉 ∥x∥∥y∥: x∈L, y∈ M x,⊥xk, y⊥ykk=1, ...,-1 where (xi,yi) ∈ L × M, i = 1,...,l, are the corresponding pairs of principal vectors. We also define sin {L, M}{colon equals}P{cyrillic}li=1 sinΘ{round}i, cos{L, M}{colon equals}P{cyrillic}li=1 cosΘ{round}i. We study relations between the principal angles and the volume of a matrix A∈Rm×n defined by vol A{colon equals}√∑det2AIJ, summing over all r×r submatrices AIJ of A. Sample results are the following generalizations of the Hadamard and Cauchy-Schwarz inequalities: 1. Let A=(A1,A2), A1∈Rn×n1l, A2∈Rn×n2m, rank A = l+m; then vol A= vol A1volA2sin{R(A1),R(A2)}. 2. Let B, C∈Rn×rr; then |det(BTC)|=volBvolCcos{R(B), R(C}. © 1992.