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, \(v_1 = 3\).