## 2.1 Definitions and matrix algebra

Definition 2.1

• An mn matrix is a rectangular grid of numbers with m rows and n columns.
• A column vector is an m✕1 matrix.
• A row vector is a 1✕n matrix.
• A square matrix is one which is mm for some m.

We typeset matrices like this: $A= \begin{pmatrix} 1 & 2 & 3 \\ 0 & -1 & \pi \end{pmatrix}, B=\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, C=\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \end{pmatrix}$ these being a 2✕3 matrix, a 3✕1 column vector, and a 3✕3 square matrix respectively.

Definition 2.2 The i, j entry of a matrix is the number in row i and column j.

For example, the 1, 2 entry of the matrix A above is 2, the 2, 1 entry is 0, and the 2, 3 entry is $$\pi$$. Very often we write $$A=(x_{ij})$$ to mean that A is a matrix whose i, j entry is $$x_{ij}$$.

If two matrices A and B are the same size (that is, they are both mn for the same m and n) then we add and subtract them by adding and subtracting each entry separately:

\begin{align*} \begin{pmatrix} 1&2\\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 0 & 1 \\ -1 & -1 \end{pmatrix} &= \begin{pmatrix} 1 & 3 \\ 2 & 3 \end{pmatrix} \\ \begin{pmatrix} 1&0 \end{pmatrix} - \begin{pmatrix} 9 & 9 \end{pmatrix} &= \begin{pmatrix} -8 & -9 \end{pmatrix} \end{align*}

We also multiply matrices by numbers one entry at a time (‘entrywise’): $2 \begin{pmatrix} 1&2&3 \\0 & 1 & 0 \end{pmatrix}= \begin{pmatrix} 2&4&6\\0&2&0 \end{pmatrix}$

This is called scalar multiplication. It satisfies some simple identities: for any matrices A and B of the same size and any number l and m, \begin{align*} (l+ m) A &= lA + m A \\ l(A+B) &= l A + l B \\ l(m A) &= (lm)A. \end{align*}

Definition 2.3 The mn zero matrix, written $$\mathbf{0}_{m\times n}$$, is the mn matrix all of whose entries are zero.

Definition 2.4 The transpose of an mn matrix A, written $$A^T$$, is the nm matrix whose i, j entry is the j, i entry of A.

To get $$A^T$$ from A you reflect A in a mirror placed along its ‘leading diagonal’: the line containing the 1, 1 entry, the 2, 2 entry, and so on. Another way to think about transpose is that the columns of A become the rows of $$A^T$$, or alternatively the rows of A become the columns of $$A^T$$.

$\begin{pmatrix} 1&2 \\ 0 & 3 \end{pmatrix}^T =\begin{pmatrix} 1 & 0 \\ 2 & 3 \end{pmatrix}, \begin{pmatrix} 1&2 \end{pmatrix}^T = \begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} a & b & c \\ d & e & f \end{pmatrix}^T = \begin{pmatrix} a & d \\ b & e \\ c & f \end{pmatrix}.$

Notice that for any matrix A we have $$(A^{T})^T = A$$.