6.1 Some definitions

Suppose we have a matrix $X$ with $n$ rows and $p$ columns; that is, $X$ is an $n\times p$ matrix. Matrices are usually denoted by an upright capital letter: $X$ .

The matrix $X^T$ is the transpose of matrix $X$ , where the rows of matrix $X$ become the corresponding columns of matrix $X^T$ . The transpose has size $p\times n$ .

Example 6.1 (Transposing) Consider the $2\times 3$ matrix $M$ , defined as $M = \left[ \begin{array}{rrr} 3 & -1 & 12\\ 2 & -4 & 0 \end{array} \right].$ Then the $3\times 2$ transpose is $M^T = \left[ \begin{array}{rr} 3 & 2 \\ -1 & -4\\ 12 & 0 \end{array} \right].$

A square matrix has the same number of rows as columns.

Example 6.2 (Square matrix) The $2\times 2$ matrix $D$ is a square matrix: $D = \left[ \begin{array}{rr} 5 & -11\\ 2.25 & 7 \end{array} \right].$

A square matrix with zeros on the off-diagonal elements is called a diagonal matrix.

Example 6.3 (Diagonal matrix) The $2\times 2$ matrix $P$ is a diagonal matrix: $P = \left[ \begin{array}{rr} -3.1 & 0\\ 0 & 2.5 \end{array} \right].$

A diagonal matrix with ones and only ones on the diagonal is called an identity matrix, sometimes written $I$ .

Example 6.4 (Identity matrix) The identity matrix of size $3\times 3$ is $\left[ \begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{array} \right],$ sometimes written $I_3$ where the subscript indicates the size of the (square) matrix.

A vector is a single-column matrix, usually denoted using bold letters. For example, consider the vector $\textbf{a} = \left[ \begin{array}{r} 1\\ 0\\ -6 \end{array} \right].$