# Tagged Questions

**1**

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**0**answers

135 views

### 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 ...

**5**

votes

**4**answers

314 views

### 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 ...

**1**

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**0**answers

61 views

### What is the time complexity of approximated SVD

Full SVD, on an m*n matrix $A$, $[U,S,V] = svd(A)$, would cost $O(m^2n + mn^2 + n^3)$ time.
But what is the time complexity if we only need the $k$ largest singular values, say, $[U_k,S_k,V_k] = ...

**0**

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**0**answers

59 views

### Cycles of Permutation Related to Rectangular Matrix Transposition

let the entries of a rectangular matrix $A\in\mathbb{C}^{m\times n}; m,n\in\mathbb{N}$ be stored in row-order in a linear vector $v$, i.e. $A_{i,j}=v_{i*m+j}$
Question:
How can the first element ...

**2**

votes

**1**answer

93 views

### Dimension independent computational complexity of singular value decomposition

Suppose $X$ is a $m \times n$ real matrix, which has only $k$ number of nonzero elements ($k \ll mn$).
Given a vector $y$, the sparsity of $X$ allows $X y$ to be computed in $O(k)$ time
which is ...

**1**

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**0**answers

100 views

### Row subset selection of matrix to optimize condition number

Given a matrix $\mathbf{A} \in \mathbb{C}^{N\times M}$ with $N \gg M$. This matrix results from a linear equation system and has a certain structure (however, I do not think that details provide any ...

**2**

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**2**answers

243 views

### Lattice reduction on an orthonormal lattice?

Suppose you are given an inner product on a vector space and given a set of linearly independent vectors, and that you have been promised that the lattice they span has an orthonormal basis. Can you ...

**12**

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**2**answers

426 views

### Optimizing the condition number

Suppose I have a set $S$ of $N$ vectors in $W=\mathbb{R}^m,$ with $N \gg n.$ I want to choose a subset $\{v_1, \dots, v_m\}$ of $S$ in such a way that the condition number of the matrix with columns ...

**2**

votes

**2**answers

544 views

### sparsity of QR decomposition

Hi, everyone!
I have a sparse $n \times n$ matrix $A$ with $nnz(A)$ denoting the number of non-zero entries in $A$. Now I use QR factorization to decompose $A$ into an orthogonal matrix $Q$ and ...

**0**

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**2**answers

105 views

### Bounding 2nd Eigenvalue of a Pseudo-Rotation-ish matrix

Let $p,q$ be arbitrary primes.
Let $N = p * q$.
Let $I$ be the $N * N$ identity matrix.
Let $R$ be the $N * N$ matrix defined as follows:
$R[x_0 * p + y_0, x_1 * p + y_1]=1$ if and only if $x_0+1 ...

**1**

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**0**answers

445 views

### 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 $\{-1,0,1\} = \square[0] = F[0]$)
...

**14**

votes

**3**answers

979 views

### 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 ...

**3**

votes

**2**answers

326 views

### 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 ...

**0**

votes

**1**answer

333 views

### 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 ...

**0**

votes

**0**answers

506 views

### 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}),\\\
...

**8**

votes

**1**answer

850 views

### 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 ...

**4**

votes

**2**answers

350 views

### 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 ...

**11**

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**0**answers

565 views

### Regular languages of matrices and their generating functions

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 ...

**11**

votes

**2**answers

2k views

### How to compute the rank of a matrix?

Okay, that's a misleading title. This is a somewhat subtler problem than undergraduate linear algebra, although I suspect there's still an easy answer. But I couldn't resist :D.
Here's the actual ...