# Questions tagged [linear-algebra]

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### Solving for the matrix $W$ in an equation involving $W \cdot W^{T}$

Having large matrices, $W$ (the unknown) and $M$ (known), is it possible to solve for $W$ in this equation $$W \cdot W^{T} = M,$$ where $M$ can have negative entries.
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### Calculating eigenvalue gap of a torus graph

Consider a two-dimensional grid with wrap-around edges (a doughnut-shaped graph). I need to calculate the second-largest eigenvalue of the adjacency matrix. Is there a faster way of computing it for ...
281 views

### Find a binary matrix so that no vector from {-1,0,1}^n is in its kernel

Given integers $n,m$, I want to find a $m \times n$ binary matrix $X$ such that there does not exist any non-zero vector $y \in \{-1,0,1\}^n$ with $Xy=0$ (all operations performed over $\mathbb{Z}$). ...
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### Generalized operators for programming languages

After asking this question on stackoverflow, it has changed slightly. Is there a way to represent a grammar as a basis for a vector space and represent a program as an object that lives in that ...
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### What are some applications of computing the permanent of a matrix?

What are some applications that require computing the permanent of a matrix? One application I know of is related to graph theory and matchings. Apparently, the number of perfect matchings of a ...
719 views

### Comparing sets of vectors

If $u,v \in \mathbb{R}^d$ are two $d$-dimensional vectors, say that $u\le v$ if $u_i \le v_i$ for all $i=1,\dots,d$. In other words, comparisons on vectors will be pointwise. Let $S,T$ be two ...
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### Solving system of linear inequalities

I am trying to solve a system of inequalities in the following form: $\ x_i - x_j \leq w$ I know these inequalities can be solved using Bellman-Ford algorithm. ...
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### Can you complete a basis in polynomial time?

Here is the problem: we are given vectors $v_1, \ldots, v_k$ lying in $\mathbb{R}^n$ which are orthogonal. We assume that the entries of $v_i$ are rational, with numerator and denominator taking $K$ ...
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### Counting solutions to system of linear equations modulo prime

I have implemented Gaussian elimination for solving system of linear equations in the field of modulo prime remainders. If there is a pivot equal to zero I assume the system has no solution but how to ...
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### What is the complexity of this matrix transposition?

I'm working on some exercises regarding graph theory and complexity. Now I'm asked to give an algorithm that computes a transposed graph of $G$, $G^T$ given the adjacency matrix of $G$. So basically ...
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### LU decomposition with pivoting

I have to solve system of linear algebraic equations $AX=B$, where $A$ is a two-dimensional matrix with all elements of main diagonal equal to zero. How to solve this problem? Iterational methods are ...
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### Machine Learning: how to correctly calculate gradient descent for simple linear problem

So, I was trying to learn machine learning, and, after watching a couple of Andrew Ng's lectures decided to try and write a simple piece of code to determine what someone's salary would be based on ...
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### Maximum feasible subsystem problem (MaxFS) in 2 variables

Topic: The maximum feasible subsystem problem, which is generally NP-hard . Question: Are there special algorithms in case of only 2 variables (2D linear constraints)? The problem seems to be a ...
I have rank-deficient matrix $M \in \mathbb{R}^{n\times m}$ with $\text{rank}(M) = k$ and I want to find a rank factorization $M = PQ$ with $P \in \mathbb{R}^{n \times k}$ and $Q \in \mathbb{R}^{k \... 1answer 165 views ### Time - Complexity Convex Optimization and Eigen Decomposition Say I had the choice of choosing one out of the following two optimization problems which I could use to solve my problem. Which choice is the fastest? How much of a trade-off would it be? Is the ... 1answer 494 views ### Probabilistic test of matrix multiplication with one-sided error Given three matrices$A, B,C \in \mathbb{Z}^{n \times n}$we want to test whether$AB \neq C$. Assume that the arithmetic operations$+$and$-$take constant time when applied to numbers from$\...
Consider a system of linear equations $Ax=0$, where $A$ is a $n\times n$ matrix with rational entries. Assume that the rank of $A$ is $<n$. What is the complexiy to check whether it has a solution \$...