Summation Convention, Dummy Indices

Consider the sum

$$s= a_1x_1 + a_2x_2 + \ldots + a_nx_n\ .$$ (2.1.1)

We can write it in a compact form as

$$s = \sum^n_{i=1}a_ix_i = \sum^n_{j=1} a_jx_j = \sum^n_{m=1}a_mx_m\ .$$ (2.1.2)

It is obvious that the index $i,\ j$ or $m$ in eqn. (2.1.2) is dummy in the sense that the sum is independent of the letter used. This is analogous to the dummy variable in an integral of a function over a finite interval.

$$I = \int^b_a f(x)dx = \int^b_a f(y) dy = \int^b_a f(t)dt\ .$$

We can further simplify the writing of eqn. (2.1.2) by adopting the following convention, sometimes known as Einstein's summation convention. Whenever an index is repeated once in the same term, it implies summation over the specified range of the index.

Using the summation convention, eqn. (2.1.2) shortens to $s = a_ix_i = a_jx_j = a_mx_m$.

Note that expressions such as $a_ib_ix_i$ are not defined within this convention. That is, an index should never be repeated more than once in the same term for the summation convention to be used. Therefore, an expression of the form

$$\sum^n_{i=1} a_ib_ix_i$$

must retain its summation sign.

In the following, unless otherwise specified, we shall always take $n$ to be 3 so that, for example,

$$\begin{split}&a_ix_i = a_mx_m = a_1x_1 + a_2x_2 + a_3x_3\ ,\\ &a_{ii} = a_{mm} = a_{11} + a_{22} + a_{33}\ .\end{split}$$

The summation convention can obviously be used to express a double sum, a triple sum, etc. For example, we can write

$$\sum^3_{i=1} \sum^3_{j=1} a_{ij} x_ix_j$$

simply as $a_{ij}x_ix_j$. This expression equals the sum of nine terms:

$$a_{ij}x_ix_j =\ a_{i1}x_ix_1 + a_{i2}x_ix_2 + a_{i3}x_ix_3\ ,$$

$$=\ a_{11}x_1x_1 + a_{21}x_2x_1 + a_{31}x_3x_1$$

$$+ a_{12}x_1x_2 + a_{22}x_2x_2 + a_{32}x_3x_2$$

$$+ a_{13}x_1x_3 + a_{23}x_2x_3 + a_{33}x_3x_3\ .$$

Similarly, the triple sum $$\sum^3_{i=1}\ \sum^3_{j=1}\ \sum^3_{k=1}\ a_ib_jc_kx_ix_jx_k$$ will simply be written as $a_ib_jc_kx_ix_jx_k$, and it represents the sum of 27 terms.

We emphasize again that expressions such as $a_{ii}x_ix_jx_j$ or $a_{ij}x_ix_jx_ix_j$ are not defined in the summation convention.

Exercise: Given

$$[a_{ij}] = \left[\begin{array}{ccc}1 & 0 & 2\\ 0 & 1 & 2\\ 3 & 0 & 3\end{array}\right].$$

Evaluate (a) $a_{ii}$, (b) $a_{ij}a_{ij}$, and (c) $a_{jk}a_{kj}$.