# eigenvalues Published articles for eigenvalues. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## Adventure Games and Eigenvalues DevFeed: [Adventure Games and Eigenvalues]() Original publisher: [Read original article]() Author: Evan Miller Published: 2017-04-23T10:10:00Z Content type: tutorial Language: en Sources: [Evan Miller]() Topics: [Mathematics](), [Algorithms, Complexity](), [Formal verification](), [Graphs]() Tags: [eigenvalues](), [formal-verification](), [graph-theory](), [matrix](), [state]() ### AI overview This article explains how Markov theory and matrix manipulation can prove the absence of dead ends in a small adventure game and quantify the risk of reaching invalid states. It compares this approach with formal verification and graph-based route-finding algorithms, while noting that all approaches become impractical for large state spaces. ### Source excerpt Identifying and quantifying dead ends, the bane of all adventure game players: Adventure Games and Eigenvalues ## A Spectral Analysis of Moore Graphs DevFeed: [A Spectral Analysis of Moore Graphs]() Original publisher: [Read original article]() Published: 2016-11-03T08:00:14Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Graphs]() Tags: [adjacency-matrix](), [eigenvalues](), [eigenvectors](), [graph](), [graphs](), [math](), [matrix](), [moore-graph](), [orthogonality](), [regular](), [spectral-graph-theory](), [trace](), [vertex]() ### AI overview This mathematical article analyzes Moore graphs of girth 5 using the eigenvalues of their adjacency matrices. It derives the minimum vertex count and shows that the degree must be one of 3, 7, or 57. ### Source excerpt For fixed integers $ r > 0$, and odd $ g$, a Moore graph is an $ r$-regular graph of girth $ g$ which has the minimum number of vertices $ n$ among all such graphs with the same regularity and girth. (Recall, A the girth of a graph is the length of its shortest cycle, and it's regular if all its vertices have the same degree) Problem (Hoffman-Singleton): Find a useful constraint on the relationship between $ n$ and $ r$ for Moore graphs of girth $ 5$ and degree $ r$. ## Principal Component Analysis DevFeed: [Principal Component Analysis]() Original publisher: [Read original article]() Published: 2012-06-28T12:08:44Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Data analysis](), [NumPy](), [Covariance](), [Python]() Tags: [covariance](), [data-analysis](), [eigenvalues](), [mathematics](), [principal-component-analysis](), [programming](), [python]() ### AI overview This tutorial explains principal component analysis as a way to reduce a dataset's dimensions by identifying directions of greatest variability. It outlines a Python and NumPy implementation that centers data, computes a covariance matrix, and obtains and sorts eigenvalues and principal components. ### Source excerpt Problem: Reduce the dimension of a data set, translating each data point into a representation that captures the "most important" features. Solution: in Python import numpy def principalComponents(matrix): # Columns of matrix correspond to data points, rows to dimensions. deviationMatrix = (matrix.T - numpy.mean(matrix, axis=1)).T covarianceMatrix = numpy.cov(deviationMatrix) eigenvalues, principalComponents = numpy.linalg.eig(covarianceMatrix) # sort the principal components in decreasing order of corresponding eigenvalue indexList = numpy.argsort(-eigenvalues) eigenvalues = eigenvalues[indexList] principalComponents = principalComponents[:, indexList] return eigenvalues, principalComponents Discussion: The problem of reducing the dimension of a dataset in a meaningful way shows up all over modern data analysis. ## Row Reduction Over A Field DevFeed: [Row Reduction Over A Field]() Original publisher: [Read original article]() Published: 2011-12-30T15:36:28Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Computing](), [Optimization]() Tags: [eigenvalues](), [eigenvectors](), [field](), [linear-algebra](), [linear-maps](), [mathematics](), [matrices](), [optimization](), [programming](), [python](), [row-reduction]() ### AI overview This tutorial introduces row reduction over a field through matrices representing linear maps between finite-dimensional vector spaces. It explains row equivalence and describes how suitable matrix forms help determine kernels, images, dimensions, eigenvalues, and eigenvectors, with applications to persistent homology and optimization problems. ### Source excerpt We're quite eager to get to applications of algebraic topology to things like machine learning (in particular, persistent homology). Even though there's a massive amount of theory behind it (and we do plan to cover some of the theory), a lot of the actual computations boil down to working with matrices. Of course, this means we're in the land of linear algebra; for a refresher on the terminology, see our primers on linear algebra. ## Google's Page Rank--The Final Product DevFeed: [Google's Page Rank--The Final Product]() Original publisher: [Read original article]() Published: 2011-06-20T22:07:41Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Google](), [Matrix](), [math](), [Web]() Tags: [computational-complexity](), [eigenvalues](), [eigenvectors](), [google](), [mathematics](), [matrix](), [page-rank](), [probability-theory](), [programming](), [pseudocode](), [ranking](), [search-engine](), [web]() ### AI overview This article explains how PageRank addresses dangling nodes and non-unique rankings. It introduces positive column-stochastic matrices and the Perron-Frobenius Theorem, then uses random web surfing as a probability-based intuition for webpage importance. ### Source excerpt Dangling Nodes and Non-Uniqueness Recall where we left off last time. Given a web $ W$ with no dangling nodes, the link matrix for $ W$ has 1 as an eigenvalue, and if the corresponding eigenspace has dimension 1, then any associated eigenvector gives a ranking of the pages in $ W$ which is consistent with our goals. The first problem is that if there is a dangling node, our link matrix has a column of all zeros, and is no longer column-stochastic. ## Linear Algebra--A Primer DevFeed: [Linear Algebra--A Primer]() Original publisher: [Read original article]() Published: 2011-06-19T18:39:40Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Matrix](), [Graphs]() Tags: [adjacency-matrix](), [algebra](), [eigenvalues](), [eigenvectors](), [graph](), [history](), [linear-algebra](), [linear-independence](), [linear-maps](), [mathematics](), [matrices](), [matrix](), [primer](), [transformation](), [vector](), [vector-spaces]() ### AI overview This primer introduces linear algebra through its historical origins in solving systems of linear equations. It explains determinants, matrices, vector algebra, and linear transformations, and shows how matrices can model graphs and compute path counts. ### Source excerpt Story Time Linear algebra was founded around the same time as Calculus (think Leibniz, circa 1700) solely for the purpose of solving general systems of linear equations. The coefficients of a system were written in a grid form, with rows corresponding to equations and columns to the unknown variables. Using a computational tool called the determinant (an awkward, but computable formula involving only the coefficients of the equations in a system), researchers were able to solve these systems, opening a world of information about the positions of celestial bodies and large-scale measurements (of geodesic arcs) on the surface of the earth. ## Google's PageRank--A First Attempt DevFeed: [Google's PageRank--A First Attempt]() Original publisher: [Read original article]() Published: 2011-06-18T18:05:20Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Graphs](), [graph theory](), [Web](), [Algorithm](), [Internet](), [structure]() Tags: [algorithm](), [analysis](), [eigenvalues](), [eigenvectors](), [google](), [graph](), [graph-theory](), [internet](), [mathematics](), [page-rank](), [ranking](), [search-engine](), [structure](), [vertex](), [web]() ### AI overview This post models the Web as a directed graph and introduces PageRank-style importance scoring for web pages. It first considers ranking pages by incoming-link counts, then explains why treating every link as equally valuable is inadequate and motivates weighting links by the importance of the linking page. ### Source excerpt The Web as a Graph The goal of this post is to assign an "importance score" $ x_i \in [0,1]$ to each of a set of web pages indexed $ v_i$ in a way that consistently captures our idea of which websites are likely to be important. But before we can extract information from the structure of the internet, we need to have a mathematical description of that structure. Enter graph theory.