Abstract
Let cos{
L,
M} ≔
Π
i = 1
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 {1, …,
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∈
J
(A)
cos
2{R(A),
R
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
J
(A) ≔ {I ∈ Q
r, m :
rank A
I∗ = r}
and
J
(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
volG =
vol A
†
cos{T, R(A
T)}
cos {S, N(A
T)}
.