5
$\begingroup$

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 $O(n\times m)$?

$\endgroup$
5
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.