Types of Matrices

Matrices come in several standard shapes, each with its own name and use.

A rectangular matrix has an unequal number of rows and columns (order $m \times n$ with $m \neq n$), while a square matrix has exactly the same number of rows as columns — an order $n \times n$ matrix is simply called a square matrix of order $n$. A matrix with just a single row is a row matrix (order $1 \times n$), and one with a single column is a column matrix (order $m \times 1$).

Within square matrices, several special forms are worth naming individually. A diagonal matrix is a square matrix in which every non-diagonal element is zero — only the entries $a_{11}, a_{22}, \dots, a_{nn}$ may be non-zero. A scalar matrix goes one step further: it is a diagonal matrix whose diagonal entries are all equal to each other. Pushed to its simplest case, an identity matrix (or unit matrix) is a scalar matrix whose diagonal entries are all $1$; it is denoted $I_n$ for order $n$, and plays the same role among matrices that the number $1$ plays among ordinary numbers.

At the other extreme, a zero matrix (or null matrix), denoted $O$, has every element equal to $0$, regardless of its order. …