# Table Interpretation Of Matrices

##### 410 words — categories: linear-algebra

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

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