# Questions tagged [linear-algebra]

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### What parts of linear algebra are used in computer science?

I've been reading Linear Algebra and its Applications to help understand computer science material (mainly machine learning), but I'm concerned that a lot of the information isn't useful to CS. For ...
2k views

### Efficient computation of Kronecker product

Given matrices $A \in \mathbb{C}^{n_1,m_1}, B \in \mathbb{C}^{n_2,m_2}$ a naive way to computer the Kronecker product would be as such: $M = \operatorname{zeros}(n_1n_2,m_1m_2)$ (initialize an empty ...
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### What is the exact definition of a dictionary in the concept of linear algebra and computer science?

I am reading this paper for my master thesis and trying to understand every concept in it. The following sentence confuses me as to what is the exact definition of a dictionary in this concept except ...
445 views

### Implementing the Schur decomposition of a matrix

I'm trying do implement the Schur decomposition of a matrix, but I can't find any good articles for the theory. Could someone share one?
277 views

### Generate algorithmically all grid points inside a hypercube

$\def\R{\mathbb{R}}\def\Z{\mathbb{Z}}\def\n#1{\|#1\|_\infty}$The problem comes directly from computational mathematics, and can be stated as follows: Given a regular matrix $M\in\R^{d\times d}$, find ...
338 views

### Partially diagonalizing a matrix

Suppose I have an $n\times n$ matrix. I would like to keep an $m\times m$ subblock undiagonized, while keeping the rest diagonized. Is there any algorithm for this? More precisely, I mean using ...
287 views

### Complexity of Pythagorean triples

We define a Pythagorean triple as a triple $\langle a,b,c\rangle$ such that $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$. In order to avoid duplicates, we say that a triple $\langle a,b,c\rangle$ is legit ...
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### Proof of Strong Duality Via Farkas Lemma

I am trying to prove what is often titled the strong duality theorem. There is a hint in the book that I'm following, and I want to follow the method they have outlined for me. I will outline the ...
368 views

### What algorithms exist for solving natural number linear systems?

I'm looking at the following problem: Given $n$-dimensional vectors of natural numbers $v_1, \ldots, v_m$ and some input vector $u$, is $u$ a linear combination of the $v_i$'s with natural number ...
697 views

### How to use different size features in SVM?

I want to train a support vector machine with some features. The problem is, one of the features is 1-dimensional (only an angle) and the other is an LBP Histogram, an 58-dimensional vector. ...
336 views

### Does spectral graph theory say anything about graph isomorphism

Is there research or are there results that discuss graph isomorphism in the context of spectral graph theory? Two known theorems of spectral graph theory are: Two graphs are called isospectral or ...
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### Subspace clustering with random transformation

One approach for clustering a high dimensional dataset is to use linear transformation, and the most common approaches are PCA and random projection (where random projection arises from the Johnson-...
755 views

### Why linear transformation can improve classification accuracy when the dimensionality of data is high?

Let $X$ be an $m\times n$ ($m$: number of records, and $n$: number of attributes) dataset. When the number of attributes $n$ is large and the dataset $X$ is noisy, classification gets more ...
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### Complexity of finding the pseudoinverse matrix

How many arithmetic operations are required to find a Moore–Penrose pseudoinverse matrix of a arbitrary field? If the matrix is invertible and complex valued, then it's just the inverse. Finding ...
Let $M$ be a square matrix with entries that are $0$ or $1$ and let $v$ be a vector with values that are also $0$ or $1$. If we are given $M$ and $y = Mv$, we can computer $v$ if $M$ is non-singular. ...
I want to code a solver for nonsingular systems of $N$ linear equations in $N$ unknowns (say up to $N=100$) with an asymmetric Toeplitz matrix. I know that the Levinson algorithm can solve it in time \$...