# Help on properties of symmetric matrices

• Nov 28th 2009, 04:06 AM
morhpeus19
Help on properties of symmetric matrices
Hi,

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

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

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

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

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

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

Thanks,