I'm reading a financial paper and my mind's drawing a blank on this following issue. Could you try help (please apologize my notation, I'm a Latex first-timer):

Say, \Sigma = \sigma\sigma^T where  \Sigma is a nxn real, symmetric variance-covariance matrix and  \sigma is a nxm matrix

Let, \lambda_i and a_i be the eigenvalues and respective eigenvectors \{1 \leq i \leq m\} and
\Sigma = ADA^T where A is a nxm orthogonal matrix comprising of those eigenvectors and D is a mxm diagonal matrix comprising of the corresponding eigenvalues

A=\left[\begin{array}{cccccc}a_1&a_2&..&a_i&..&a_m\end{arr  ay}\right] , a_i = \{a_{ki}, 1 \leq k  \leq n\}

I'll denote each element of matrix \sigma by \sigma_{ik}, \{1 \leq i \leq n, 1 \leq k \leq m\}

Let dz(t) be a mx1 vector of orthogonal processes

Then, prove that \sum^m_{k=1}\sigma_{ik}dz_k(t) = \sum^m_{k=1}\sqrt{\lambda_k}a_{ik}dz_k(t)