# Tagged Questions

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### Is finding a single vector in the null space as difficult as discovering the whole null space?

Let $A \in \mathbb R^{k\times n}$ be a matrix of rank $k$, where $k \ll n$. One can use Gaussian eliminations to discover $\operatorname{null}(A)$ at the cost of $O(nk^2)$. My question is: Is the ...
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### NP-hard problems in linear algebra and real analysis [closed]

I am curious about NP-hard problems in linear algebra and real analysis. An example in linear algebra would be the calculation of the permanent. I would thus like to collect in this thread a list of ...
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### Representing vertices of a cube using linear combination of tensor product of smaller cubes

Let $n,N \in \mathbb{N}$ with $N \ge n^{2}$. Let $F[i] = \square[i]$ refer to the cube which has vertices from $\{-1,0,1\}^{n^{i}}$ ($n^{i}$ tuple of alphabets from $\{-1,0,1\} = \square[0] = F[0]$) ...
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### symmetric integer matrices

Suppose I have a symmetric positive definite matrix $M$ with integer entries. I want to decide whether $M = A A^t,$ with $A$ likewise integral. I assume that decision problem is NP-complete, as is the ...
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### Finding the solution to b = Ax that minimizes the Hamming weight (everything over the field F_2).

Is there an efficient algorithm for finding the solution $x$ of $b = Ax$ that minimizes the Hamming weight of $x$, where $A$ is a nxm-matrix over the field $\mathbb{F}_2$ ("integer matrix modulo ...
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### What is the relationship between singular value decomposition and solving linear systems?

It is known that solving systems of linear equations is reducible to SVD in a straightforward way; if you want to solve $\mathbf{Ax}=\mathbf{b}$, then you can perform SVD on $\mathbf{A}$ and minimize ...
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### How to calculate inner product with some set of (0 or 1,0 or 1,0 or 1, … )vectors in the fastest way ?

Given: some vector $R=(r_1...r_l)$ - real numbers, and a set of distinct vectors with $0$ or $1$ coordinates \begin{array}{c} V_1=(c_{1,1} ... c_{1,l}),\\\ V_2=(c_{2,1} ... c_{2,l}),\\\ ...
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### An algorithm to find non-trivial linear dependencies

This question is inspired by another MO question about special stratifications of equivariant Grassmannians, that turned out to be a problem of computing non-trivial circuits in a vector matroid. To ...
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### A Boolean function that is not constant on affine subspaces of large enough dimension

I'm interested in an explicit Boolean function $f \colon \{0,1\}^n \rightarrow \{0,1\}$ with the following property: if $f$ is constant on some affine subspace of $\{0,1\}^n$, then the dimension of ...
My question is somewhat related to this question. Let us fix natural numbers $k$ and $C$. Let $A$ be an automaton whose alphabet consists of $k\times k$ matrices with integer coefficients of ...