## 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 3.1 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 2.10 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 2.11 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 5.1 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 $\textbf{a} = \left[ \begin{array}{r} 1\\ 0\\ -6 \end{array} \right].$