## Abstract

Let cos{L, M} := Π_{i = l} cos θ_{i} denote the product of the cosines of the principal angles {θ_{i}} between the subspaces L and M. The direction cosines of an r-dimensional subspace L are the (^{n}_{r}) numbers {cos{L, ℝ^{n}_{J}}: J ∈ Q _{r, n}}, where Q_{r, n} := the set of increasing sequences of r elements from {l, . . . , n}, and ℝ^{n}_{J} := {x = (x_{k}) ∈ ℝ^{n} : x_{k} = 0 for k ∉ J}. The basic decomposition of a linear operator A : ℝ^{n} → ℝ^{m}, with rank A = r > 0, is A = ∑_{I ∈ ℐ(A)} ∑_{J ∈ Script J sign(A)} cos^{2}{R(A), ℝy^{m}_{l}}; cos^{2}{R(A^{T}), ℝ^{n}_{J}}B_{IJ}, a convex combination of nonsingular linear operators B_{IJ} : ℝ^{n}_{J} → ℝ^{n}_{I}. Here ℐ(A) := {I ∈ Q_{r, m} : rank A_{I*} = r} and Script J sign(A) := {J ∈ Q_{r, 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(A^{T})} cos{S, N(A^{T})}.

Original language | English (US) |
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Pages (from-to) | 71-81 |

Number of pages | 11 |

Journal | Linear Algebra and Its Applications |

Volume | 237-238 |

DOIs | |

State | Published - Apr 1996 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

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