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.
Channel
At this time we can state the remaining additional condition required that a decomposition be unique. The index space I has to be partitioned into exactly two parts, say I′ and I″; i.e.,
I′ ∪ I″ = I(8)
I′ ∩ I″ = φ,
such that
dim(X′) = dim(X″),(9)
where
| X′ = | ⨂Xᵢ(10) |
| I′ | |
| X″ = | ⨂Xᵢ. |
| I″ |
(If dim (X) is odd, then we have to cheat a little by putting in an extra random dummy dimension.) And then the decomposition of the space