# Questions tagged [linear-algebra]

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33 views

### Scalability of factor-solve vs. pseudoinverse + product

I’ve always read advice and warnings about the poor scalability and time spent trying to invert a matrix and why it’s better to solve a system of equations whenever the inverted matrix will be used ...
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### What is the bit complexity of Gaussian eliminaton over $\Bbb F_q$?

Given matrix $M\in\Bbb F_q^{n\times n}$ with rank $r$ what is the complexity of converting to row-echelon form? Is it $O(n^3\log q)$ or $O(n^3q)$ bit operations? Technically $O(n^3)$ row ...
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### Testing whether a set of integers can be written as a combination of module basis elements

Input We are given a set of basis elements, $\ v_1$,$\ v_2$ ,...,$\ v_n$ of a $\mathbb Z^m$- module and a multiset of integers $\ B :=$ {$\ b_1, ..., b_m$} Desired Output Return true if there ...
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### How many iterations of Lanczos bidiagonalization are required in order to obtain the first k singular values/vectors of a matrix?

I am trying to implement a fast SVD algorithm for obtaining the first $k$ singular values/vectors of an $M\times N$ matrix ($k < \min(M,N)$) using the following 2-step process: 1) bidiagonalize ...
55 views

### Computational Complexity of Integer Linear Program [with Fixed number of 'Pure Constants']

This is a follow up to the previous Question: Conditions for Linear Diophantine Equations to always have a solution It was established in the above's answer that obtaining or testing for the ...
131 views

### Conditions for Linear Diophantine Equations to always have a solution

Given a set of $n$ linear equations in $v$ integer variables, where $v > n$, we can say that this system of equations will always have an integer solution. Over $\mathbb N$, is there any such ...
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### small size and small depth circuit for set intersections

Input: Given sets $S_i \subseteq \{1,2,3,4,\cdots,n\}$ for $1 \leq i \leq n$. Output: sets intersection with restriction (pick first set $S_1$. If $a \in S_1$ such that $a$ is the least element then ...
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### Efficient algorithm to check if a vector exists in the span of a subset of vectors

I have a binary vector $v \in \mathbb{F}_2^m$ and a set of binary vectors $Q=\{q_1,q_2,\dots,q_n\}$ each belonging to $\mathbb{F}_2^m$ and I know that $v \in \text{span}\{Q\}$ but I want to know if ...
725 views

### Suitable choice for moderate-size square matrix multiplication?

The problem is to find $C = AB$, where $A$ and $B$ are $n \times n$ matrices that may be sparse. Let $n$ be around 1000. The elements of $A$ and $B$ are real values, though, for practicality's sake, ...
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### Permutation on matrix to fill main diagonal with non-zero values

I am currently working on some sparse non-singular matrices. One of the algorithms I use requires divisions by the elements on the main diagonal so I have to ensure that my main diagonal is filled ...
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### Is this a known question in matrix sketching?

Say one has a $D \times n$ matrix $A$ all of whose entries are non-zero. One wants a method which will look at each of the columns of $A$ one by one and create new $m \ll D$ dimensional columns and ...
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### How to construct a running kd-tree?

I have a stream of 3-tuples of type (x,y,t) where x and y are in the range ...
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### Computing Von Neumann Entropy Efficiently

The Von Neumann entropy $S$ of a density matrix $\rho$ is defined to be $S(\rho)= -\text{tr}(\rho \lg \rho)$. Equivalently, $S$ is the classical entropy of the eigenvalues $\lambda_k$ treated as ...
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### Intuitive self-contained proof of Farkas' Lemma

I've been studying the proof of Farkas' Lemma, and given my rather fuzzy memory of Linear Algebra, am having some trouble with it. One version of Farkas' lemma states: For any convex cone generated ...
764 views

### It is possible to implement a *greater than* function using only addition, substractions and multiplications?

All values are from a finite field $Z_t$. I want to write a function greater than like this $GT(x,y) = \begin{cases} 1, & \text{if } x > y, \\ 0, & \text{otherwise}. \end{cases}$ ...
209 views

### Solving systems of linear equations over semirings

So I have come across an issue where it would be very nice to solve systems of linear equations over semirings but I have no clue how to do that. Over a field I would use Gaussian elimination but I'm ...