# Balanced Mixing for Target Covariance

Let $\mathbf{X}$ be a vector of i.i.d. random variables. Let $\mathbf{C}$ be a desired covariance matrix for which I like to determine mixing matrix $\mathbf{A}$ such that

$$var(\mathbf{A}\mathbf{X}) = \mathbf{A}^T\mathbf{A} = \mathbf{C}.$$

Now, this could be solved by the Cholesky or eigenvalue decomposition depending on the definiteness of $\mathbf{C}$. I am, however, interested in a balanced mixing matrix, i.e.,

$$| a_{ij} | = |a_{ji}|$$

The idea is that the resulting random variables are as mixed as possible. In contrary, if $\mathbf{A}$ is triangular (as from the Cholesky decomposition) at least one random variable is identical to the original which is undesired.

A numerical example: Let $$\mathbf{C} = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$$ be the desired covariance matrix. Possible mixing matrices are $$\mathbf{A}_1 = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix} / \sqrt{3} \textrm{ and } \mathbf{A}_2 = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$ where $\mathbf{A}_1^T \mathbf{A}_1 = \mathbf{C}$ and $\mathbf{A}_2^T \mathbf{A}_2 = \mathbf{C}$ and $\mathbf{A}_1$ is more balanced than $\mathbf{A}_2$.

What is a systematic method to find a balanced mixing matrix (such as $\mathbf{A}_1$) instead of an unbalanced mixing matrix (such as $\mathbf{A}_2$)?

Let's say you can distill your balance criterion into an objective function $f(A)$. Once we find one viable matrix $A_0$, for example by Cholesky, all other matrices can be derived as $QA_0$ where $Q$ is orthogonal such that $Q^TQ=I$. Therefore your problem reduces to finding an orthogonal matrix that maximizes $f(QA_0)$. Depending on your function $f$, there might even be closed form expression for $Q$.