# Chapter 6 Matrix algebra, very briefly

Matrix algebra is well-suited for ecology, because most (if not all) *data sets* we work with are in a *matrix* format.

## 6.1 Data sets *are* matrices

Ecological data tables are obtained as object-observations or sampling units, and are often recorded as this:

Objects | \(y_1\) | \(y_2\) | \(\dots\) | \(y_n\) |
---|---|---|---|---|

\(x_1\) | \(y_{1,1}\) | \(y_{1,2}\) | \(\dots\) | \(y_{1,n}\) |

\(x_2\) | \(y_{2,1}\) | \(y_{2,2}\) | \(\dots\) | \(y_{2,n}\) |

\(\vdots\) | \(\vdots\) | \(\vdots\) | \(\ddots\) | \(\vdots\) |

\(x_m\) | \(y_{m,1}\) | \(y_{m,2}\) | \(\dots\) | \(y_{m,n}\) |

where \(x_m\) is the sampling unit \(m\); and \(y_n\) is the ecological descripor that can be, for example, species present in a sampling unit, locality, or a chemical variable.

The same ecological data table can be represented in *matrix notation* like this:

\[Y = [y_{m,n}] = \begin{bmatrix} y_{1,1} & y_{1,2} & \cdots & y_{1,n} \\ y_{2,1} & y_{2,2} & \cdots & y_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ y_{m,1} & y_{m,2} & \cdots & y_{m,n} \end{bmatrix}\]

where lowercase letters indicate *elements*, and the subscript letters indicate the *position of these elements* in the matrix (and in the table!).

Moreover, any subset of a matrix can be recognized.

We can subset *a row matrix*, as below:

\[\begin{bmatrix} y_{1,1} & y_{1,2} & \cdots & y_{1,n} \\ \end{bmatrix}\]

We can also subset *a column matrix*, as below:

\[\begin{bmatrix} y_{1,1} \\ y_{2,2} \\ \vdots \\ y_{m,2} \end{bmatrix}\]

## 6.2 Association matrices

Two important matrices can be derived from the ecological data matrix: the * association matrix among objects* and the

*.*

**association matrix among descriptors**Using the data from our matrix \(Y\),

\[ Y = \begin{array}{cc} \begin{array}{ccc} x_1 \rightarrow\\ x_2 \rightarrow\\ \vdots \\ x_m \rightarrow\\ \end{array} & \begin{bmatrix} y_{1,1} & y_{1,2} & \cdots & y_{1,n} \\ y_{2,1} & y_{2,2} & \cdots & y_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ y_{m,1} & y_{m,2} & \cdots & y_{m,n} \end{bmatrix} \end{array} \]

one can examine the relationship between the first two objects:

\[x_1 \rightarrow \begin{bmatrix} y_{1,1} & y_{1,2} & \cdots & y_{1,n} \\ \end{bmatrix} \]

\[x_2 \rightarrow \begin{bmatrix} y_{2,1} & y_{2,2} & \cdots & y_{2,n} \\ \end{bmatrix} \]

and obtain \(a_{1,2}\).

We can populate the association matrix \(A_{n,n}\) with the relationships between all objects from \(Y\):

\[A_{n,n} = \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n,1} & a_{n,2} & \cdots & a_{n,n} \end{bmatrix}\]

Because \(A_{n,n}\) has the same number of rows and columns, it is denoted a *square matrix*.

Therefore, \(A_{n,n}\) has \(n^2\) elements.

We can also obtain the relationship between the first two descriptors of \(Y\), \(y_1\) and \(y_2\):

\[\begin{bmatrix} y_{1,2} \\ y_{2,2} \\ \vdots \\ y_{m,2} \end{bmatrix}\]

\[\begin{bmatrix} y_{1,1} \\ y_{2,1} \\ \vdots \\ y_{m,1} \end{bmatrix}\]

and store it in \(a_{1,2}\).

We can populate the association matrix \(A_{m,m}\) with the relationships between all descriptors from \(Y\):

\[A_{m,m} = \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,m} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,m} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,m} \end{bmatrix}\]

This \(A_{m,m}\) is a *square matrix*, and it has \(m^2\) elements.

These matrices, \(A_{n,n}\) and \(A_{m,m}\), are the basis of ** Q-mode** and

**analyses in ecology.**

*R-mode*** R-mode** constitutes of analyzing the association between descriptors or species, while

**analyzes the association between OTUs, objects or sites.**

*Q-mode*