Product cosines of angles between subspaces

Jianming Miao, Adi Ben-Israel

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


Let cos{L, M} := Πi = l cos θi denote the product of the cosines of the principal angles {θi} between the subspaces L and M. The direction cosines of an r-dimensional subspace L are the (nr) numbers {cos{L, ℝnJ}: J ∈ Q r, n}, where Qr, n := the set of increasing sequences of r elements from {l, . . . , n}, and ℝnJ := {x = (xk) ∈ ℝn : xk = 0 for k ∉ J}. The basic decomposition of a linear operator A : ℝn → ℝm, with rank A = r > 0, is A = ∑I ∈ ℐ(A)J ∈ Script J sign(A) cos2{R(A), ℝyml}; cos2{R(AT), ℝnJ}BIJ, a convex combination of nonsingular linear operators BIJ : ℝnJ → ℝnI. Here ℐ(A) := {I ∈ Qr, m : rank AI* = r} and Script J sign(A) := {J ∈ Qr, n : rank A*J = r}. The product cosines are related to the matrix volume, defined as the product of its nonzero singular values. The Moore-Penrose inverse A is characterized as having the minimal volume among all {1, 2}-inverses of A. Indeed, if G is a {1, 2}-inverse of A, with range R(G) = T and null space N(G) = S, then vol G = vol A †/cos{T, R(AT)} cos{S, N(AT)}.

Original languageEnglish (US)
Pages (from-to)71-81
Number of pages11
JournalLinear Algebra and Its Applications
StatePublished - Apr 1996
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics


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