Hi!,

I´ve tried to carry out the calculations of equations 15.4.19 and 15.4.20 according to the scheme introducced along the Section 15.4.1 (computing the eq. 15.4.12).

What I have found for equation 15.4.19 is

\sum_{i=0}^{M-1}(U_{ji}/w_i)^2

that differs in the U from the one written in the book with V.

The covariance expression I have found is more complicated than 15.4.20. I don´t know if I´m missing some cancellation from U'U=UU'=1 and/or V'V=1...

Has someone done these steps? Any suggestion?

Thanks in advance!!

OK, here's a hint -- you'll have to fill in all the indices to turn it into a real proof. ^T means transpose and <> means expectation.

a = V diag(1/w_j) U^T b
Cov(a,a) = <aa^T>-<a><a^T>
= V diag(1/w_j) U^T (<bb^T>-<b><b^T>)U diag(1/w_j) V^T

But the b's have unit variance (eq. 15.4.5) and are assumed independent, so the thing in the middle is the identity matrix. Now the U^T U is also the identity matrix, and the result is basically equation (15.4.20), with (15.4.19) as the diagonal elements.

Hope this helps.