<story><title>Viewing Matrices and Probability as Graphs</title><url>https://www.math3ma.com/blog/matrices-probability-graphs</url></story><parent_chain></parent_chain><comment><author>edflsafoiewq</author><text>Re viewing matrices as graphs&lt;p&gt;Under the usual matrices-as-linear-maps interpretation, a nxm matrix is a linear map from the free vector space on m elements to the free vector space on n elements, M : Vect(Y) -&amp;gt; Vect(X), where X and Y are as in the article.&lt;p&gt;For a regular map of finite sets Y -&amp;gt; X, we can draw a picture showing how each element of Y goes to an element of X by drawing a bunch of dots for Y and a bunch of dots for X with an arrow from each dot in Y to its image in X. This is a directed bipartite graph where each dot in Y is the source of exactly one edge.&lt;p&gt;How is a linear map, Vect(Y) -&amp;gt; Vect(X), different? Each element of Y goes not to a single element of X, but to a linear combination of elements of X. If y goes to a x1 + b x2, we can think of this as two arrows out of y, each one now being weighted, one arrow to x1 with weight a and one to x2 with weight b (considering edges with weight 0 to be non-existent if you like). This is the graph shown in the article.&lt;p&gt;This interpretation shows why matrix multiplication is path composition. It&amp;#x27;s exactly analogous to how for the map of finite sets, you find the image of z by first following its arrow to y in Y, then following &lt;i&gt;&lt;/i&gt;its&lt;i&gt;&lt;/i&gt; arrow to x in X. Here you first follow z to all its images in Y, then you follow those to all their images in X. Relation composition is only a special case.&lt;p&gt;And of course the connection between XxY-&amp;gt;R and Vect(Y)-&amp;gt;Vect(X) is the usual correspondence between a rank (1,1) tensor and a linear map.&lt;p&gt;Re probabilities: this corresponds to imagining a random process that transforms elements of Y into elements of X where y turns into x with probability p(x,y). The probability that you end up with x (the marginal probability) is the probability that any dot in Y transformed into x, ie. sum over all the arrows that wind up at x.</text></comment>
<story><title>Viewing Matrices and Probability as Graphs</title><url>https://www.math3ma.com/blog/matrices-probability-graphs</url></story><parent_chain></parent_chain><comment><author>ironSkillet</author><text>If you like this idea, also check out &lt;a href=&quot;https:&amp;#x2F;&amp;#x2F;graphicallinearalgebra.net&quot; rel=&quot;nofollow&quot;&gt;https:&amp;#x2F;&amp;#x2F;graphicallinearalgebra.net&lt;/a&gt;.&lt;p&gt;It also describes fundamental linear algebra operations in terms of graphs, but with a slightly different flavor, and much more detail.</text></comment>