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Suppose we have two matrices $A_{m\times n}$ and $B_{m\times m}$. Such that $B$ is a symmetric positive definite matrix. Is it possible to compute main diganoal of $A^TBA$ in $O(n\times m)$?

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Not unless $\omega = 2$. Take $B = \operatorname{id}$, $A = \begin{bmatrix}X & Y\end{bmatrix}$. You can extract $XY$ from $A^TBA$.

UPDATE: I missed the main diagonal part of the question. Even computing the main diagonal is as hard as matrix multiplication: denote $f(A, B) = \operatorname{tr}(A^TBA) = \sum_{i,j,k} a_{ij} b_{ik} a_{kj}$. The derivatives $\frac{\partial f(A, B)}{\partial a_{ij}}$ form the matrix $BA + A^TB$, and we can apply the Baur-Strassen theorem.

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