|Θ(⨀Xᵢ)| = sin⁻¹_e_^{-R(⨀Xᵢ)} (5)
I
and the norm
|X| = κ(2π_e_)⁻¹ᐟ² _e_^{(HX)}. (6)
Now, if[19] a statistically independent basis; _i.e._, one for which
κ
R(⨀Xᵢ) ≡ constant, (7)
I
can be provided in terms of one-dimensional components; _i.e._, none
of them can be decomposed further, then it is just the usual problem
of diagonalization of a symmetric matrix by means of a congruence
transformation to provide an orthogonal coordinate system. Furthermore,
for uniqueness, we arrange the spectrum in decreasing order. Then,
by means of the Radon Nikodym theorem applied to each of these
one-dimensional axes, the probability distribution may be made; _e.g._,
Gaussian, if desired. Thus, we obtain the promised orthogonal Euclidean
space.
[19] This “if” is the catch that makes all methods of metrization of a
space of dimensionality higher than one impractical, except the method
of successive projections upon unit spheres centered at the center of
gravity. The method of using that nilpotent projection operator is
described in the companion paper(see footnote page 65).