# Questions tagged [linear-algebra]

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### Finding the common part of two systems of linear equations modulo 2

Given two systems of linear equations modulo 2, is there a method to find the common part (the equations which are true in both case) ? For example... Matrix 1: a⊕b=1 and c⊕d=0 and y=1 and e=0 Matrix ...
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### Let M be a k × n random matrix with iid entries such that

$M$ is a $k × n$ random matrix with iid entries such that $P(m_{i,j} = +1) = P(m_{i,j} = −1) = 0.5.$ Let $k = O({1\over \epsilon^l})$ for some constant $l$. $v ∈ R_n$ is a fixed vector. Does a ...
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### Complexity of a decision problem: system of linear equations over finite field with restricted solutions

I have a system of linear equations over a finite field $\mathbb F_p \cong \mathbb Z_p$, and I'm interested in the decision problem of whether there exists a solution where all of the variables $x_i$ ...
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### Why is the weight matrix diagonal in weighted least squares regression?

I was going through the theory for weighted least-squares fitting and I understood its basic underlying concepts, but I couldn't understand why exactly do we keep the weights as a diagonal matrix ...
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### Better way to decide if a set is a pure simplicial complex

Setup I am trying to write a function that determines if a set of vertices, edges and faces is a pure simplicial complex. A pure simplicial complex is a set where all facets have the same degree, a ...
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### Randomized Assignment Problem

Given $x_1,...,x_n,y_1,...,y_n\in \mathbb{R}^d$ find a permutation matrix $P\in\mathbb{S}_d$ that minimizes $\sum_{ij}P_{ij}|x_i-y_j|$. This is an assignment problem and can be solved in $O(n^3+n^2d)$ ...
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### N-dimensional generalization of map and reduce?

Is there any conceptual generalization of higher-order functions like map and reduce but for N-dimensional objects (e.g. arrays or tensors)? For mapping, I guess it would be a point-wise ...
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### Seeking guidance on what to read for Feasibility Binary IP with ''almost total unimodular'' (-1, 0, 1)-Coefficient Matrix and No Obj Function

I am working on an algorithm in graph theory which I wish to prove it's polynomiality/NP-hardness. I am investigating a binary variable (0, 1) integer program which has the coefficient matrix ...
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### How does the sweep line algorithm check for intersection using vector cross product?

I am trying my best to understand the sweep-line algorithm to find line intersections. I have understood most of the intuition except how it is calculating the intersection between 2 line segments ...
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### When does Gaussian elimination solve exact 1-in-3 SAT?

Terms: A literal is a variable or its negation. A clause is a set of literals. An exact 3-in-1 clause is satisfied if an assignment of values to variables results in exactly 1 ...
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### Geometric median of two disjoint sets of points lies on line between their respective medians

I was working on a problem about geometric medians and I had an idea for a divide and conquer solution, but it would only work if a set of points, when split into two disjoint sets, and those sets ...
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### How to quickly solve a linear equation for 7000 times?

I need to solve a linear equation Ax=b for 7000 times (A is sparse and complex square matrix), and at each time only 4 elements (A(i,k), A(i,m), A(j,k) and A(j,m)) are changed while all other elements ...
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### How to make sure matrix completion can generate a matrix with values in expected range?

I am doing a matrix completion project. Assume that I have an incomplete matrix like ...
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### What is the fastest algorithm to establish whether a linear system in $\mathbb{R}$ has a solution?

I know the best algorithm to solve a linear system in $\mathbb{R}$ with $n$ variables is Coppersmith-Winograd's algorithm, which has a complexity of $$O\left(n^{2.376}\right).$$ How much easier is ...
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### Transforming a byte with a subset of a small, fixed set of values and xor into any other value

If I have some collection of bits, -- a byte, say -- of arbitrary value then I can transform it into some other value by means of exclusive-oring it with a subset of (in this case) eight fixed values, ...
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### Smallest Circuit for Square of Sparse Symmetric Matrix

I have an n by n symmetric matrix, and I would like to compute its square in as small a circuit complexity as possible. It's sparse: there are sqrt(n) nonzero entries in each row/column, so the input ...