$m\times n$ matrices

An $m\times n$ matrix (``over $F$'') is a rectangular array or table of elements of $F$ arranged with $m$ rows and $n$ columns. It is usually written:

$$A=\left( \begin{array}{ccccc} a_{11}&a_{12}&...&a_{1n}\\ ... ... &\vdots \\ a_{m1}&a_{m2}&...&a_{mn} \end{array} \right).$$

The $(i,j)^{th}$ entry of $A$ is $a_{ij}$. The $i$th row of $A$ is

$$\left[ \begin{array}{rrrr} a_{i1} & a_{i2} & \cdots & a_{in} \end{array} \right] \;\;\;\;\;\;(1\leq i\leq m)$$

The $j$th column of $A$ is

$$\left[ \begin{array}{c} a_{1j} \\ a_{2j} \\ \vdots \\ a_{mj} \end{array} \right] \;\;\;\;\;\;(1\leq j\leq n)$$

A matrix having as many rows as it has columns ($m=n$) is called a square matrix. The entries $a_{ii}$ of an $m\times n$ matrix $A=(a_{ij})$ are called the diagonal entries, the entries $a_{ij}$ with $i>j$ are called the lower diagonal entries, and the entries $a_{ij}$ with $i<j$ are called the upper diagonal entries. An $m\times n$ matrix $A=(a_{ij})$ all of whose lower diagonal entries are zero is called an upper triangular matrix. This terminology is logical if the matrix is a square matrix but both the matrices below are called upper triangular

$$\begin{array}{cc} \left( \begin{array}{cccc} 1& 2&3&4\\ ... ... 0&0 &0&10 \\ 0&0 &0&0 \end{array} \right) \end{array}$$

whether they look triangular of not! A similar definition holds for lower triangular matrices. The square $n\times n$ matrix with $1$'s on the diagonal and 0's elsewhere,

$$\left( \begin{array}{cccc} 1& &...&0\\ 0&\ddots& &\vdots\\ \vdots& & &0 \\ 0&\vdots &0&1 \end{array} \right),$$

is called the $n\times n$ identity matrix and denoted $I$ or $I_n$. This is both upper triangular and lower triangular. (In general, any square matrix which is both upper triangular and lower triangular is called a diagonal matrix.) A square $n\times n$ matrix with exactly one $1$ in each row and each column, and 0's elsewhere, is called an $n\times n$ permutation matrix. The identity $I_n$ is a permutation matrix. We shall discuss these types of matrices in detail in the next chapter. A square $n\times n$ matrix with exactly one non-zero entry in each row and each column, and 0's elsewhere, is called an $n\times n$ monomial matrix. We shall discuss these types of matrices later in the book. They have many properties similar to permutation matrices. Monomial matrices occur in the explicit description of the ``isometry class of a linear code'' (in §5.5 below).