But don’t be scared, because we’re going to break it down for you like a warm apple pie (without the math).

To set the stage: what is Einstein notation? It’s basically shorthand for writing out equations with matrices and vectors. Instead of using fancy symbols and subscripts, we just write everything in plain old regular letters. For example, instead of this:

Aij * xj = yi

We can use Einstein notation to simplify it like so:

Aix * xx = yi

See? Much easier on the eyes (and much less intimidating). But let’s dive a little deeper into what exactly is going on here.

In this equation, Aij represents our matrix of coefficients, and we’re multiplying it by a vector xj to get another vector yi. The i and j subscripts are just telling us which row or column we’re dealing with for example, if we have the following 2×2 matrix:

A = [1 2;
3 4]

Then Aij would look like this:

A[i][j]

So if i=1 and j=2, that would be our bottom right element (which is a 4). But in Einstein notation, we don’t need to write out all those subscripts. Instead, we just use the same letter for both the row and column indices this is called summation convention.

So let’s say we have a vector x with three elements:

x = [2; 4; 6]

We can multiply it by our matrix A using Einstein notation like so:

Aix * xx = yi

This means that for each row i in the matrix, we’re multiplying all of its corresponding column elements (which are represented by j) with their respective vector element (also represented by j), and then adding up those products. The result is a new vector yi which contains our transformed values.

It may not be as exciting as apple pie, but at least now you can impress your friends with your newfound knowledge (or scare them away).