While a vector is a list of numbers that depend on only one index, the most direct interpretation of a matrix is a table of numbers that depend on two indices, \(i\), and \(j\). In this case, the rows and columns of the matrix usually have some simple interpretation. Here are some examples.

Images. A black and white image with \(M \times N\) pixels is naturally represented as an \(M \times N\) matrix.

Rainfall Data. An \(m \times n\) matrix \(A\) gives the rainfall at \(m\) different locations on \(n\) consecutive days, so \(A_{42}\) is the rainfall at location 4 on day 2.

Asset Returns. A \(T \times n\) matrix \(R\) gives the returns of a collection of \(n\) assets (the universe of assets) over \(T\) periods (it’s a time series), with \(R_{ij}\) giving the return of asset \(j\) in period \(i\). So \(R_{12, 7} = -0.03\) means that asset 7 had a 3% loss in period 12.

Supplier Prices. An \(m \times n\) matrix \(P\) gives the prices of \(n\) different goods from \(m\) different suppliers. \(P_{ij}\) is the price that supplier \(i\) changes for good \(j\).

Customer Purchase History. An \(n \times N\) matrix \(P\) can be used to store a set of \(N\) customers’ purchase histories of \(n\) products, items, or services, over some period.

Matrix Representation of a Collection of Vectors

If \(x_i, \cdots, x_N\) are \(n\)-vectors that give the \(n\) feature values for each of \(N\) objects, we can collect them into one \(n \times N\) matrix often called a data matrix or feature matrix.

\[X = [ \ x_1 \ x_2 \ \cdots \ x_N \ ]\]

Matrix Representation of a Relation or Graph

Suppose we have \(n\) objects labeled \(1, \cdots, n\). A relation \(\mathcal{R}\) on the set of objects \({1, \cdots, n}\) is a subset of ordered pairs of objects. \(\mathcal{R}\) could represent a preference relation among \(n\) possible products or choices, with \((i, j) \in \mathcal{R}\) meaning that choice \(i\) is preferred to choice \(j\). A relation can also be viewed as a directed graph, with nodes labeled \(1, \cdots, n\) and a directed edge from \(j\) to \(i\) for each \((i, j) \in \mathcal{R}\). This is typically drawn as a graph, with arrows indicating the direction of the edge. Normally, a relation is represented by the \(n \times n\) matrix \(A\) with

\[A_{ij} = \begin{cases} 1 & (i, j) \in \mathcal{R} \\ 0 & (i, j) \notin \mathcal{R} \end{cases} \]