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.