on angles between subspaces of inner product spaces - FMIPA ...

{v1, . . . , vq} are orthonormal, the area of the parallelogram spanned by projV u1 and
� q�
q� 2 projV u2 is 〈u1, vk〉 〈u2, vk〉 2 � q�
− 〈u1, vk〉〈u2, vk〉 �2 �1/2 , while the area of the
k=1
k=1
k=1
parallelogram spanned by u1 and u2 is 1.
Our result here says that, in statistical terminology, the value of cos θ is the product
of the maximum and the minimum correlations between u ∈ U and its best predictor
proj V u ∈ V . The maximum correlation is of course the cosine of the first canonical
angle, while the minimum correlation is the cosine of the second canonical angle. The
fact that the cosine of the last canonical angle is in general the minimum correlation
between u ∈ U and proj V u ∈ V is justified in [10].
2.2. The general case. The main ideas of the proof of Theorem 1 are that the
matrix M T M is diagonalizable so that its determinant is equal to the product of its
eigenvalues, and that the eigenvalues are the square of the cosines of the canonical
angles. The following statement is slightly less sharp than Theorem 1, but much easier
to prove (in the sense that there are no sophisticated arguments needed).
Theorem 3. cos θ is equal to the product of all critical values of f(u) := max 〈u, v〉,
v∈V,�v�=1
u ∈ U, �u� = 1.
Proof. For each u ∈ X, 〈u, v〉 is maximized on {v ∈ V : �v� = 1} at v = projV u
�projV u� and
max 〈u, v〉 = �projV u�. Now consider the function g := f
v∈V,�v�=1
2 , given by
g(u) = �projV u� 2 , u ∈ U, �u� = 1.
Writing u = p�
tiui, we have
i=1
proj V u =
q�
〈u, vk〉vk =
k=1
q�
k=1
and so we can view g as g(t1, . . . , tp), given by
� q� p� �
�
�
g(t1, . . . , tp) := � ti〈ui, vk〉vk
where p�
k=1
i=1
� 2
p�
ti〈ui, vk〉vk,
i=1
=
q� � p � �2 ti〈ui, vk〉 ,
t
i=1
2 i = 1. In terms of the matrix M = [〈ui, vk〉] T , one may write
k=1
i=1
g(t) = t T M T Mt = |Mt| 2 q, |t|p = 1,
where t = [t1 · · · tp] T ∈ R p and | · |p denotes the usual norm in R p .
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