# Representation of Generalized Quadratic Form of Random Variables

I am interested in finding/understanding a "good" $$L^2$$-orthogonal (i.e. uncorrelated) decomposition of a matrix-valued quadratic form. The setup is as follows:

Given a random matrix $$X$$ (tall, of size $$n\times N$$), with mean zero rows $$X_i$$ consider $$Q(X) = X'AX.$$ The covariance structure of $$X$$ can be encoded in a block matrix $$\Sigma>0$$ given as $$\Sigma = \begin{bmatrix} \Sigma_{11} & \Sigma_{12} & \dots& \Sigma_{1N}\\ \Sigma_{12} & \Sigma_{22} & \dots&\\ \vdots & \ddots& \ddots \end{bmatrix}.$$ The blocks are of size $$n$$ corresponding to the length of the vectors $$X_i$$.

Ideally, I would like to represent $$Q$$ as $$Q(X) = X'AX = \sum U'_i\Lambda_iU_i$$ Where the covariance structure of the $$U_i$$ is at least block-diagonal: $$\Lambda = \begin{bmatrix} \Lambda_1 & 0 & 0&\dots\\ 0 & \Lambda_2 & 0 &\dots\\ 0 & 0 & \ddots & \ddots \end{bmatrix}.$$ Is this possible? Also, and if so, I am interested in the relationship between the singular values of the blocks of $$\Sigma$$ and $$\Lambda$$.

I suspect something like this to be acheivable given that we can do it ordinary quadratic forms: sum of squares of dependent gaussian random variables

I guess maybe one could use a vectorized version of the argument presented there, but I am not entirely sure how to procede?