# 5.1: What is a matrix?

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To start with, a matrix is nothing more than a rectangular arrangement of a set of elements (actually, it's a bit more complicated than that, but we will return to matrices of more than two dimensions in a little bit). Rectangles have sizes that are described by the number of rows of elements and columns of elements that they contain. A "3 by 6" matrix has three rows and six columns; an "I by j" matrix has I rows and j columns. A matrix that has only one row is called a "row vector." A matrix that has only one column is called a "column vector.

Figure 5.1 shows a two-by-four matrix. Figure 5.2 shows a four by two matrix. Just for the moment, ignore the contents of the cells (e.g. 1,1).

Figure 5.1. Example of a "two-by-four" matrix

1,1 | 1,2 | 1,3 | 1,4 |

2,1 | 2,2 | 2,3 | 2,4 |

Figure 5.2. Example of at "four-by-two" matrix

1,1 | 1,2 |

2,1 | 2,2 |

3,1 | 3,2 |

4,1 | 4,2 |

The elements (cells) of a matrix are identified by their "addresses." Element 1,1 is the entry in the first row and first column; element 13,2 is in the 13th row and is the second element of that row. The cell addresses have been entered as matrix elements in the two examples above.

Matrices are often represented as arrays of elements surrounded by vertical lines at their left and right, or square brackets at the left and right. In web pages it's easier to use "tables" to represent matrices. Matrices can be given names; these names are usually presented as capital bold-faced letters. Social scientists using matrices to represent social networks often dispense with the mathematical conventions, and simply show their data as an array of labeled rows and columns. The labels are not really part of the matrix, but are simply for clarity of presentation. The matrix in figure 5.3 for example, is a 4 by 4 matrix, with additional labels.

Figure 5.3. Four-by-four matrix with additional row and column labels

A | B | C | D | |

A |
--- |
1 |
0 |
0 |

B |
1 |
--- |
1 |
0 |

C |
1 |
1 |
--- |
1 |

D |
0 |
0 |
1 |
--- |

The matrices used in social network analysis are frequently "square." That is, they contain the same number of rows and columns. But "rectangular" matrices are also used, as are row and column vectors. The same conventions apply to all these variations.

Occasionally, social network analysts will use a "3-dimensional" matrix. A three dimensional matrix has rows, columns, and "levels" or "slices." Each "slice" has the same rows and columns as each other slice. UCINET thinks about these more complicated 3-dimensional arrays of data as a collection of two-dimensional matrices.