Gradient of a function with respect to a non-square matrix

137 Views Asked by Bumbble Comm At 11 Apr 2026 - 6:04

Consider the following function of a matrix $W$ of size $d \times p$.

$$f(W)= aW (W^T \Sigma W)^{-1} W^Ta^T$$

Here: $a \in R^{d}$ is a constant vector, $\Sigma$ is a constant covariance matrix. I am trying to compute the gradient of $f(W)$ w.r.t $W$.

I have checked the matrix cookbook at

However, none of the formula provided on the document helps me.

Thanks in advance.