# Tag Info

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### Is this inequality involving the Frobenius norm right?

For a short fat matrix $G$ (more columns than rows), $\|AG\|_F \geq \sigma_{\min}(G)\|A\|_F \geq n \sigma_{\min}(G) \|A\|$, where $\sigma_{\min}(G)$ is the least singular value of $G$. This follows ...

### Solving a system of linear equations over the integers

To make sure I understand: you have an $m \times n$ matrix and vector in $\mathbb{Z}^n.$ Is the system a priori over- or under-determined? The former case is a little easier than the latter, but in ...

### Is it faster to compute eigenvalues or coefficients of characteristic polynomials?

With the traditional algorithms and complexity measures used in numerical linear algebra (dense real matrices, floating point computations, flop count as a complexity measure), they are both more or ...
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### Complexity of rectangular matrix multiplication

Assuming that efficient means better than the naive $O(n^{2+k})$ multiplication, let us review some possibilities. Padding. For $k > \omega-2$, just pad $A$ with $n-n^k$ zero or garbage rows, ...

### Computation time of Smith normal form in Maple

(Dense) Smith Normal Form is theoretically computable in $O(\|A\| \log \|A\| N^4\log N)$ time (Arne Storjohann, 1996). Storjohann was at Waterloo at the time, so I would not be surprised if that is ...
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### Matrix equation with Hadamard product and its own inverse involved

Removing all unnecessary parameters, we come to the equation $\Omega^{-1}=2 W\odot \Omega + B$ where $B$ is positive definite. We need to find a solution in the cone $M_+$ of positive definite ...
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### When are two binary matrices simultaneously equivalent to their transpose?

Clearly, a necessary condition is that for every word $w$ in two letters, one has $${\rm Tr}\,w(A^t,B^t)={\rm Tr}\,w(A,B).$$ Equivalently, $${\rm Tr}\,\hat w(A,B)={\rm Tr}\,w(A,B),$$ where $\hat w$ is ...
According to the man page of svds, provided by MATLAB, svds is currently based on "Augmented Lanczos Bidiagonalization Algorithm" when it comes to the top-$k$ SVD computation of a large-...