3.1 Definitions and conventions

A matrix is a collection of numbers arranged in a rectangular array of rows and columns, such as

$$\left[ \begin{array}{rr} 3 & 2 \\ 4 & -1 \\ -1 & 2 \end{array} \right]$$

A matrix with $r$ rows and $c$ columns is said to be an $r \times c$ matrix (e.g., the matrix above is a $3 \times 2$ matrix).

In the case where a matrix has just a single row or column, it is said to be a vector, such as

$$\left[ \begin{array}{r} 3 \\ -1 \end{array} \right]$$

Conventionally, vectors and matrices are denoted in lower- and upper-case boldface, respectively (e.g., $x$ is a scalar, $\mathbf{x}$ is a vector, and $\mathbf{X}$ is a matrix). In addition, vectors are taken to be column vectors – i.e., a vector of $n$ numbers is an $n \times 1$ matrix, not a $1 \times n$ matrix.

The $ij$th element of a matrix $\mathbf{M}$ is denoted by $M_{ij}$ or $(\mathbf{M})_{ij}$.

For example, letting $\mathbf{M}$ denote the above matrix, $M_{11}=3$, $(\mathbf{M})_{32}=2$, and so on. Similarly, the $j$th element of a vector $\mathbf{v}$ is denoted $v_j$; e.g., letting $\mathbf{v}$ denote the above vector, $v_1 = 3$.