# Tag Info

## Hot answers tagged matrix-product

### Evidence that matrix multiplication is not in $O(n^2\log^kn)$ time

There's an algorithm for multiplying an $N \times N^{0.172}$ matrix with an $N^{0.172} \times N$ matrix in $N^2 \operatorname{polylog}\left(N\right)$ arithmetic operations. The main identity used for ...
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### smallest circuit size using XOR gates

This is NP-hard. See: Joan Boyar, Philip Matthews, René Peralta. Logic Minimization Techniques with Applications to Cryptology. http://link.springer.com/article/10.1007/s00145-012-9124-7 The ...
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### Evidence that matrix multiplication is not in $O(n^2\log^kn)$ time

Well, one thing is I think that all the constructions we know of - and even the families of potential constructions that people have proposed (e.g., Cohn-Umans approaches, generalizations of ...
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### Matrix multiplication with transpose

No. Consider block matrices $A = \left(\begin{matrix} 0 & 0 \\ 0 & X \end{matrix}\right)$ (with symmetric $X$) and $B = \left(\begin{matrix} 0 & Y \\ 0 & 0 \end{matrix} \right)$. ...
Accepted

### Commutative matrix multiplication algorithms

In answer to the "Update": yes, for any $c$, the existence of an $O(n^c)$ non-commutative algorithm for matrix multiplication is equivalent to the existence of an $O(n^c)$ commutative algorithm for ...
• 35.9k
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### Fast Finding Main Diagonal of Matrix Multiplication

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 ...
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### Arithmetic complexity of matrix powering

It depends on the relationship between $m$ and $d$. If $m \geq 3$ is fixed, but $d$ is allowed to grow without bound, then the corresponding class of functions is exactly the same as functions with ...
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### Is the exponent in the rectangular matrix multiplication convex?

Does Lemma 3.6 of https://arxiv.org/abs/2009.10217 answer your original question of convexity of the matrix multiplication constant?
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### Low-depth arithmetic complexity of the product of $k$ matrices

I am not sure about specifically depth-three lower bounds, but there has been a lot of depth-4 (and 5) lower bounds, usually assuming other constraints as well. For instance (and without any claim of ...
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### Multiplication of circulant matrices with a diagonal matrix

This problem is essentially equivalent to matrix multiplication. Let's look at $\sum_{i=1}^{n}X_{i}RY_{i}$ first, where $X_i$ and $Y_i$ are diagonal matrices. If $A$ is the matrix with the diagonal ...
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### Is the exponent in the rectangular matrix multiplication convex?

I found an answer in the paper "Fast sparse matrix multiplication" as Theorem 2.4. The authors cite "Fast rectangular matrix multiplications and applications", so that's the ...
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Sorry if I am missing something, but isn't it always singular in your special case? The first column is identically $1$, the second column is $(\beta_1,\ldots,\beta_{n^2})^T$, and the $n+1$'th column ...
Because $\mathbf{c}' = R'\mathbf{p}$ and $R'$ contains 3 nonzero elements in each row.