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Eigenvectors and Eigenvalues explained visually

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Eigenvalues/vectors are instrumental to understanding electrical circuits, mechanical systems, ecology and even Google's PageRank algorithm. Let's see if visualization can make these ideas more intuitive. To begin, let $v$ be a vector (shown as a point) and $A$ be a matrix with columns $a_1$ and $a_2$ (shown as arrows). If we multiply $v$ by $A$, then $A$ sends $v$ to a new vector $Av$. If you can draw a line through the three points $(0,0)$, $v$ and $Av$, then $Av$ is just $v$ multiplied by a number $\lambda$; that is, $Av = \lambda v$. In this case, we call $\lambda$ an eigenvalue and $v$ an eigenvector. For example, here $(1,2)$ is an eigvector and $5$ an eigenvalue. Below, change the columns of $A$ and drag $v$ to be an eigenvector. Note three facts: First, every point on the same line as an eigenvector is an eigenvector. Those lines are eigenspaces, and each has an associated eigenvalue. Second, if you place $v$ on an eigenspace (either $s_1$ or $s_2$) with associated eigenvalue $

Eigenvectors and Eigenvalues explained visually Back Eigenvectors and Eigenvalues Explained Visually Tweet By Victor Powell and Lewis Lehe Eigenvalues/vectors are instrumental to understanding electrical circuits, mechanical systems, ecology and even Google's PageRank algorithm. Let's see if visualization can make these ideas more intuitive. To begin, let $v$ be a vector (shown as a point) and $A$ be a matrix with columns $a_1$ and $a_2$ (shown as arrows). If we multiply $v$ by $A$, then $A$ sends $v$ to a new vector $Av$. If you can draw a line through the three points $(0,0)$, $v$ and $Av$,

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