Matrices/K/Introduction/Section

A system of linear equations can easily be written with a matrix. This allows us to make the manipulations that lead to the solution of such a system without writing down the variables. Matrices are quite simple objects; however, they can represent quite different mathematical objects (e.g., a family of column vectors, a family of row vectors, a linear mapping, a table of physical interactions, a relation, a linear vector field, etc.), which one has to keep in mind in order to prevent wrong conclusions.

Definition

Let ${}K$ denote a field, and let ${}I$ and ${}J$ denote index sets. An ${}I\times J$-matrix is a mapping

$I\times J\longrightarrow K,(i,j)\longmapsto a_{ij}.$

If ${}I=\{1,\ldots ,m\}$ and ${}J=\{1,\ldots ,n\}$, then we talk about an ${}m\times n$-matrix. In this case, the matrix is usually written as

${\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}.$

We will usually restrict to this last situation.

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 ${}\operatorname {Mat} _{n}(K)$.

Two matrices ${}A,B\in \operatorname {Mat} _{m\times n}(K)$ are added by adding corresponding entries. The multiplication of a matrix ${}A$ with an element ${}r\in K$ (a scalar) is also defined entrywise, so

${\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}+{\begin{pmatrix}b_{11}&b_{12}&\ldots &b_{1n}\\b_{21}&b_{22}&\ldots &b_{2n}\\\vdots &\vdots &\ddots &\vdots \\b_{m1}&b_{m2}&\ldots &b_{mn}\end{pmatrix}}={\begin{pmatrix}a_{11}+b_{11}&a_{12}+b_{12}&\ldots &a_{1n}+b_{1n}\\a_{21}+b_{21}&a_{22}+b_{22}&\ldots &a_{2n}+b_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}+b_{m1}&a_{m2}+b_{m2}&\ldots &a_{mn}+b_{mn}\end{pmatrix}}\,$

and

${}r{\begin{pmatrix}a_{11}&a_{12}&\ldots &a_{1n}\\a_{21}&a_{22}&\ldots &a_{2n}\\\vdots &\vdots &\ddots &\vdots \\a_{m1}&a_{m2}&\ldots &a_{mn}\end{pmatrix}}={\begin{pmatrix}ra_{11}&ra_{12}&\ldots &ra_{1n}\\ra_{21}&ra_{22}&\ldots &ra_{2n}\\\vdots &\vdots &\ddots &\vdots \\ra_{m1}&ra_{m2}&\ldots &ra_{mn}\end{pmatrix}}\,.$

The multiplication of matrices is defined in the following way:

Definition

Let ${}K$ denote a field, and let ${}A$ denote an ${}m\times n$-matrix and ${}B$ an ${}n\times p$-matrix over ${}K$. Then the matrix product

$AB$

is the ${}m\times p$-matrix, whose entries are given by

${}c_{ik}=\sum _{j=1}^{n}a_{ij}b_{jk}\,.$

A matrix multiplication is only possible when the number of columns of the left-hand matrix equals the number of rows of the right-hand matrix. Just think of the scheme

${}(ROWROW){\begin{pmatrix}C\\O\\L\\U\\M\\N\end{pmatrix}}=(RC+O^{2}+WL+RU+OM+WN)\,,$

the result is an ${}1\times 1$-Matrix. In particular, one can multiply an ${}m\times n$-matrix ${}A$ with a column vector of length ${}n$ (the vector on the right), and the result is a column vector of length ${}m$. The two matrices can also be multiplied with roles interchanged,

${}{\begin{pmatrix}C\\O\\L\\U\\M\\N\end{pmatrix}}(ROWROW)={\begin{pmatrix}CR&CO&CW&CR&CO&CW\\OR&O^{2}&OW&OR&O^{2}&OW\\LR&LO&LW&LR&LO&LW\\UR&UO&UW&UR&UO&UW\\MR&MO&MW&MR&MO&MW\\NR&NO&NW&NR&NO&NW\end{pmatrix}}\,.$