In the previous post, I described a simple way to think about matrices, namely as bipartite graphs. Today I'd like to share a different way to picture matrices—one which is used not only in mathematics, but also in physics and machine learning. Here's the basic idea. An $m\times n$ matrix $M$ with real entries represents a linear map from $\mathbb{R}^n\to\mathbb{R}^m$. Such a mapping can be pictured as a node with two edges. One edge represents the input space, the other edge represents the output space.

That's it!

We can accomplish much with this simple idea. But first, a few words about the picture: To specify an $m\times n$ matrix $M$, one must specify all $mn$ entries $M_{ij}$. The index $i$ ranges from 1 to $m$—the dimension of the output space—and the index $j$ ranges from 1 to $n$—the dimension of the input space. Said differently, $i$ indexes the number of rows of $M$ and $j$ indexes the number of its columns. These indices can be included in the picture, if we like:

This idea generalizes very easily. A matrix is a two-dimensional array of numbers, while an $n$-dimensional array of numbers is called a tensor of order $n$ or an $n$-tensor. Like a matrix, an $n$-tensor can be  represented by a node with one edge for each dimension.

A number, for example, can be thought of as a zero-dimensional array, i.e. a point. It is thus a 0-tensor, which can be drawn as a node with zero edges. Likewise, a vector can be thought of as a one-dimensional array of numbers and hence a 1-tensor. It's represented by a node with one edge. A matrix is a two-dimensional array and hence 2-tensor. It's represented by a node with two edges. A 3-tensor is a three-dimensional array and hence a node with three edges, and so on.