In much of the material of related to physics, one finds it expedient to adapt the summation convention first introduced by Einstein. Let us consider first the set of linear equations
![{\displaystyle {\begin{matrix}a_{1}x+b_{1}y+c_{1}z&=&d_{1}\\a_{2}x+b_{2}y+c_{2}z&=&d_{2}\\a_{3}x+b_{3}y+c_{3}z&=&d_{3}\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6755c2ef2ecfc413431c92fd0fd6bbf760947edc)
We shall find it to our advantage to set
,
,
. The superscripts do not denote powers but are simply a means for distinguishing between the three quantities
,
, and
. One immediate advantage is obvious. If we were dealing with 29 variables, it would be foolish to use 29 different letters, one letter for each variable. The single letter
with a set of superscripts ranging from 1 to 29 would suffice to yield the 29 variables, written
,
,
,
,
. Our reason for using superscripts rather than subscripts will soon become evident. Equations (1) can now be written
![{\displaystyle {\begin{matrix}a_{1}x^{1}+b_{1}x^{2}+c_{1}x^{3}&=&d_{1}\\a_{2}x^{1}+b_{2}x^{2}+c_{2}x^{3}&=&d_{2}\\a_{3}x^{1}+b_{3}x^{2}+c_{3}x^{3}&=&d_{3}\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6fd58bb98927abf3463b9d52598384b807c1b1)
Equations (2) still leave something to be desired, for if there were 29 such equations, our patience would be exhausted in trying to deal with the coefficients of
,
,
,
,
. Let us note that in (2) the coefficients of
,
,
may be expressed by the matrix
![{\displaystyle \left({\begin{matrix}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{matrix}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a606bd6990112ae56fc69d7422f900f21abe041)
By defining
,
,
,
,
,
,
,
,
, the matrix (3) becomes
![{\displaystyle \left({\begin{matrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{matrix}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdc22b68cd840f971d257a431d232d331083eca6)
One advantage is immediately evident. The single element
lies in the i th row and j th column of the matrix (4). Equations (1) can now be written
![{\displaystyle {\begin{matrix}a_{11}x^{1}+a_{12}x^{2}+a_{13}x^{3}&=&d_{1}\\a_{21}x^{1}+a_{22}x^{2}+a_{33}x^{3}&=&d_{2}\\a_{31}x^{1}+a_{32}x^{2}+a_{33}x^{3}&=&d_{3}\end{matrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0ddab2efe0833dfb1716375ca84cefc010bc86c)
Using the familiar summation notation of mathematics, we rewrite (5) as
![{\displaystyle \sum _{r=1}^{3}a_{1r}x^{r}=d_{1}\qquad \sum _{r=1}^{3}a_{2r}x^{r}=d_{2}\qquad \sum _{r=1}^{3}a_{3r}x^{r}=d_{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec8e7c28ef85535dce1ed32fec74540304b7011)
or in even shorter form
![{\displaystyle \sum _{r=1}^{3}a_{ir}x^{r}=d_{i}\qquad i=1,2,3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c79377b3ae5ea1b9c5644aa694a6bb102e5967ad)
The system of equations
![{\displaystyle \sum _{r=1}^{3}a_{ir}x^{r}=d_{i}\qquad i=1,2,3,\dots ,n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf70b293029ba224ea2e3623a37a7d4a575b1b2a)
represents
linear equations.
Einstein noticed that it was excessive to carry along the
sign in (8). we may rewrite (8) as
![{\displaystyle a_{ir}x^{r}=d_{i}\qquad i=1,2,3,\dots ,n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6da2dd1debe9da298c916e0bd11b0db5ed7e5a90)
provided it is understood that whenever an index occurs exactly once both as a subscript and superscript a summation is indicated for this index over its full range of definition. In (9) the index
occurs both as a subscript (in
) and as a superscript (in
), so that we sum on
from
to
. In a four-dimensional spacetime (
) summation indices range from 1 to 4. The index of summation is a dummy index since the final result is independent of the letter used. We can write
![{\displaystyle a_{ir}x^{r}\equiv a_{ij}x^{j}\equiv a_{i\alpha }x^{\alpha }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ca3a188a5c2f74d4e943b6bcf7ebc6e3b60fdd3)
We may also write (9) as
![{\displaystyle a_{r}^{i}x^{r}=d^{i}\qquad i=1,2,3,\dots ,n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2031c886e304b0c51ef23d10397623adb09cc56)
where the element
belongs to the i th row and j th column of the matrix
![{\displaystyle \left({\begin{matrix}a_{1}^{1}&a_{2}^{1}&\dots &a_{n}^{1}\\a_{1}^{2}&a_{2}^{2}&\dots &a_{n}^{2}\\\dots &\dots &\dots &\dots \\a_{1}^{n}&a_{2}^{n}&\dots &a_{n}^{n}\end{matrix}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a491d894072b984091076f6467ac70f931adfce4)