Question: I am reading Jackson's 'Classical Electrodynamics' (3rd Edition), and as an exercise I was trying to go through equation 6.142 on page 268. I was wondering if he got this definition of the matrix multiplication incorrect? Jackson writes:

$$\sum\_\alpha a\_{\alpha\beta}a\_{\alpha \gamma}=\delta\_{\beta \gamma}$$

But, when explicitly expanding the summation for a 2x2 matrix I get an error for the $a\_{11}$ entry (all entries as a matter of fact):

$$a\_{11} = a\_{11}a\_{11}+a\_{21}a\_{21}$$

When it should read:

$$a\_{11} = a\_{11}a\_{11}+a\_{12}a\_{21}$$

This 'error' can be corrected using the matrix multiplication summation equation:

$$\sum\_\alpha a\_{\beta\alpha}a\_{\alpha \gamma}=\delta\_{\beta \gamma}$$

My Einstein summation abilities are limited and very rusty, so I just thought I'd check in to make sure I am understanding things correctly, or not.

Answer: 
That equation doesn't define general matrix multiplication. For general matrix multiplication, the formula is

$$C= AB: C\_{ij} = \sum\_k A\_{ik} B\_{kj}$$

Translating his summation into a matrix multiplication would mean

$$\delta = a^T a$$
as you can deduce from the correct matrix multiplication formula I just gave.

If $\delta$ happens to be the Kronecker-$\delta$, then what this equation expresses is that matrix $a$ is *orthogonal*. I don't have a copy of Jackson at hand to check for the context though.

It's not an *error* per se, it's just that the equation you provide is, indeed, not the general definition of matrix multiplication, but rather a special case of multiplying a matrix with its own transpose.
More context which confirms [Lagerbaer's answer](https://physics.stackexchange.com/a/304714/44126), since my Jackson is at hand:

> 
> A rotation in three dimensions is a linear transformation of the coordinates of a point such that the sum of the squares of the coordinates remains invariant.
>  Such a transformation is called an orthogonal transformation.
>  The transformed coordinates $x\_\alpha'$ are given in terms of the original coordinates $x\_\beta$ by
>  $$
> x\_\alpha'= \sum\_\beta a\_{\alpha\beta}x\_\beta \tag{6.141}
> $$
>  The requirement to have $(\mathbf{x'})^2 = (\mathbf x)^2$ restricts the real transformation coefficients $a\_{\alpha\beta}$ to be orthogonal,
>  $$
> \sum\_\alpha a\_{\alpha\beta} a\_{\alpha\gamma} = \delta\_{\beta\gamma} \tag{6.142}
> $$
>  The inverse transformation has $(a^{-1})\_{\alpha\beta} = a\_{\beta\alpha}$ and the square of the determinant of the matrix $(a)$ is equal to unity. ...
> 
> 
> 

So what you have here is not a general expression for matrix multiplication, but a restriction on this particular matrix in order to give it the symmetries it needs to represent rotations.