Basic Mathematics For Quantum Mechanics — Eigenvectors

Let’s consider a matrix A and a vector v. When we multiply A and v together we get back another vector w. So

Normally w is completely different to v. However, sometimes w is multiple of v. What I mean by this is that there is some scalar, s, that when I multiply v by it I get w. So

For example

In the cases where w is a scalar multiple of v we get:

And we call v an eigenvector of A with eigenvalue s

Let’s look at an example. Consider

So v is an eigenvector of A with eigenvalue -3

Eigenvectors and values are extremely useful in all sorts of scenarios but this is all we need to know for now.