Matrix/Definition/Explanations/Remark

For every ${}i\in I$, the family ${}a_{ij}$ , ${}j\in J$, is called the ${}i$-th row of the matrix, which is usually written as a row tuple (or row vector)

$(a_{i1},a_{i2},\ldots ,a_{in}).$

For every ${}j\in J$, the family ${}a_{ij}$ , ${}i\in I$, is called the ${}j$-th column of the matrix, usually written as a column tuple (or column vector)

${\begin{pmatrix}a_{1j}\\a_{2j}\\\vdots \\a_{mj}\end{pmatrix}}.$

The elements ${}a_{ij}$ are called the entries of the matrix. For ${}a_{ij}$, the number ${}i$ is called the row index, and ${}j$ is called the column index of the entry. The position of the entry ${}a_{ij}$ is where the ${}i$-th row meets the ${}j$-th column. A matrix with ${}m=n$ is called a square matrix. An ${}m\times 1$-matrix is simply a column tuple (or column vector) of length ${}m$, and an ${}1\times n$-matrix is simply a row tuple (or row vector) of length ${}n$. The set of all matrices with ${}m$ rows and ${}n$ columns (and with entries in ${}K$) is denoted by ${}\operatorname {Mat} _{m\times n}(K)$; in case ${}m=n$ we also write ${}\operatorname {Mat} _{n}(K)$.