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.


2 Answers 2


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, 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.


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