n. Proof 1:

An element in SY is given by, where n = number of inputs, and m = number of outputs

Yij = Σr=1n Σk=1n Xrk Hir Hjr*         i=1:m         j=1:m

A diagonal element is given by:

Yii = Σr=1n Σk=1n Xrk Hir Hir* = Σr=1n Xrr |Hir|2         i=1:m

If SX is diagonal, Xrk = 0, if r ≠ k, then:

Yii = Σr=1n Xrr |Hir|2         i=1:m

This can be written as a set of linear equations:

Y = HX

Which can be solved for X as: X = H-1Y