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

\[Av = w\]

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

\[w = sv\]

For example

\[\begin{bmatrix}4 \\ 2\end{bmatrix} = 2\begin{bmatrix}2 \\ 1\end{bmatrix}\]

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

\[Av = sv\]

And we call v an eigenvector of A with eigenvalue s

Let’s look at an example. Consider

\[A = \begin{bmatrix}-5 & 2 \\ -9 & 6\end{bmatrix}\] \[v = \begin{bmatrix}1 \\ 1\end{bmatrix}\] \[Av = \begin{bmatrix}-3 \\ -3\end{bmatrix} = -3\begin{bmatrix}1 \\ 1\end{bmatrix}\]

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.