Definition 1

Let $I, K, H$ be three sets; a matrix of type $(I, K)$ with elements in $H$ (or a matrix of type $(I, K)$ over $H$) is any family $M = (m_{i\kappa})_{(i, \kappa) \in I \times K}$ of elements of $H$ whose indexing set is the product $I \times K$. For all $i \in I$, the family $(m_{i\kappa})_{\kappa \in K}$ is called the row of $M$ of index $i$; for all $\kappa \in K$, the family $(m_{i\kappa})_{i \in I}$ is called the column of $M$ of index $\kappa$.

If $I$ (resp. $K$) is finite, $M$ is said to be a matrix with a finite number of rows (resp. columns). The set of matrices of type $(I, K)$ over $H$ is identified with the product $HI^{I \times K}$.

The names "row" and "column" arise from the fact that, in the case where $I$ and $K$ are intervals $[1, p], [1, q]$ of $\mathbf{N}$, the elements of the matrix are envisaged as set out in a rectangular array with $p$ rows (arranged horizontally) and $q$ columns (arranged vertically):

When $p$ and $q$ are explicit integers sufficiently small for it to be practicable, it is a convention that the above array is a symbol effectively denoting the matrix in question; this notation enables us to dispense with the use of indices, it being understood that the indices of an element are determined by its place in the array; for example, when we speak of the matrix

we mean the matrix $(m_{ij})_{1 \leq i < 2, 1 \leq j < 3}$ such that

Instead of matrix of type $([1, p], [1, q])$, we also say matrix of type $(p, q)$, or matrix with $p$ rows and $q$ columns, if no confusion arises; the set of matrices of type $(p, q)$ over $H$ is sometimes denoted by $\mathbf{M}_{p, q}(H)$.

Every matrix over $H$ for which one of the indexing sets $I, K$ is empty is identical with the empty family of elements of $H$; it is also called the empty matrix. When $I = \{i_0\}$ (resp. $K = \{k_0\}$) is a set consisting of a single element, $M$ is called a row matrix (resp. column matrix) and the row (resp. column) index can then be suppressed in the notation; when $I$ and $K$ are both sets with one element, a matrix of type $(I, K)$ is often identified with the unique element in this matrix.

A subfamily $M' = (m_{\iota \kappa})_{(\iota, \kappa) \in J \times L}$ of a matrix $M = (m_{\iota \kappa})_{(\iota, \kappa) \in I \times K}$, whose indexing set is the product of a subset $J$ of $I$ and a subset $L$ of $K$, is called a submatrix of the matrix $M$; it is said to be obtained by suppressing in $M$ the rows of index $\iota \notin J$ and the columns of index $\kappa \notin L$; conversely, $M$ is said to be obtained by bordering $M'$ with the rows of index $\iota \notin J$ and the columns of index $\kappa \notin L$.