$C=\sum_{i=1}^{n} Z(X_{i}-\mu)Z^{T}$ . Find $Z$ if $X$ follows a bivariate normal distribution

Question

Let $C=VDV^T$ where $D$ is a diagonal matrix and $V^T V=I$. Now I have an expression of the form $$C=\sum_{i=1}^n Z(X_i-\mu)(X_i-\mu)^{T}Z^T$$ Here $C$ is known and $X_i$ are known, moreover $X_i$ follows a standard bivariate gaussian distribution ($X \in \mathcal{N}(0,I_{2\times 2})$). We need to find the value of $Z.$ $C$ is a $2 \times 2$ matrix and $Z$ is also a $2 \times 2 $ matrix. How should I proceed? Everything except $Z$ is known or can be calculated. We just want to write $Z$ in terms of all the other variables.

Answer 1

Let $S=\sum_i (X_i-\mu)(X_i-\mu)^T$ have the eigenvector (or spectral) decomposition $S=Q\Lambda Q^T$, where $QQ^T=I$. Now you want $$ VDV^T = ZSZ^T = Z Q \Lambda Q^T Z^T. $$ Then $Z=VD^{1/2}\Lambda^{-1/2}Q^T$ does the trick, where $\Lambda^{-1/2}$ and $D^{1/2}$ are the obvious diagonal matrices.