# 6 Geometric Duality

In chapter Data Matrix we talked about how a data table can be mathematically treated as a data matrix: typically as an array of individuals and variables. In this chapter we take a further step that should let you adopt a **geometrical mindset**. More specifically, you will learn how to think and look at any data matrix from a geometric standpoint. This is an incredible insightful concept which some authors refer to as the *duality* of a data matrix.

## 6.1 Matrix Perspectives

It’s very enlightening to think of a data matrix as viewed from the glass of Geometry. The key idea is to think of the data in a matrix as elements living in a multidimensional space. Actually, we can regard a data matrix from two apparently different perspectives that, in reality, are intimately connected: the *rows perspective* and the *columns perspective*. In order to explain these perspectives, let me use the following diagram of a data matrix \(\mathbf{X}\) with \(n\) rows and \(p\) columns, with \(x_{ij}\) representing the element in the \(i\)-th row and \(j\)-th column.

When we look at a data matrix from the *columns* perpective what we are doing is focusing on the \(p\) variables. In a similar way, when looking at a data matrix from its *rows* perspective, we are focusing on the \(n\) individuals. This double perspective or **duality** for short, is like the two sides of the same coin.

### 6.1.1 Rows Space

We know that human vision is limited to three-dimensions, but pretend that you had superpowers that let you visualize a space with any number of dimensions.

Because each row of the data matrix has \(p\) elements, we can regard individuals as objects that live in a \(p\)-dimensional space. For visualization purposes, think of each variable as playing the role of a dimension associated to a given axis in this space; likewise, consider each of the \(n\) individuals as being depicted as a point (or particle) in such space, like in the following diagram:

In the figure above, even though I’m showing only three axes, you should pretend that you are visualizing a \(p\)-dimensional space (imaging that there are \(p\) axes). Each point in this scape corresponds to a single individual, and they all form what you can call a *cloud of points*.

### 6.1.2 Columns Space

We can do the same visual exercise with the columns of a data matrix. Since each variable has \(n\) elements, we can regard the set of \(p\) variables as objects that live in an \(n\)-dimensional space. For convention purposes, and in order to distinguish them from the individuals, we use an arrow (or vector) to graphically represent each variable:

Analogously to the rows space and its cloud of individuals, you should also pretend that the image above is displaying an \(n\)-dimensional space with a bunch of blue arrows pointing in various directions.

### 6.1.3 Toy Example

To make things less abstract, let’s consider a toy \(2 \times 2\) data matrix of *weight* and *height* values measured on two individuals Leia and Luke. Here’s the code in R to create a matrix `X`

for this example:

```
# toy matrix
X <- matrix(c(150, 172, 49, 77), nrow = 2, ncol = 2)
rownames(X) <- c("Leia", "Luke")
colnames(X) <- c("weight", "height")
X
weight height
Leia 150 49
Luke 172 77
```

**Space of individuals**: If you look at the data in `X`

from the individuals standpoint, each of them can be depicted as a dot in the 2-dimensional space of variables `weight`

and `height`

:

**Space of variables**: If you look at the data in `X`

from the variables standpoint, each of them can be depicted as a vector in the 2-dimensional space spanned by `Leia`

and `Luke`

:

### 6.1.4 Dual Space

The following figure illustrates the dual perspective of a data matrix. The standard convention is to represent **individuals as points** in a \(p\)-dimensional space; and in turn represent **variables as vectors** in an \(n\)-dimensional space.