31. Eigenvectors and eigenvalues

31. Eigenvectors and eigenvalues

Eigenvectors and eigenvalues

Given a complicated matrix, it can be difficult to determine salient information just by looking at the array of numbers. For example, the matrix: $M=\begin{pmatrix}\frac{1}{3}&\frac{1}{3}+\frac{1}{\sqrt{3}}&-\frac{1}{3}+\frac% {1}{\sqrt{3}}\\ \frac{1}{3}-\frac{1}{\sqrt{3}}&\frac{1}{3}&-\frac{1}{3}-\frac{1}{\sqrt{3}}\\ -\frac{1}{3}-\frac{1}{\sqrt{3}}&-\frac{1}{3}+\frac{1}{\sqrt{3}}&\frac{1}{3}% \end{pmatrix}$ describes a 90 degree rotation around $\begin{pmatrix}1\\ 1\\ -1\end{pmatrix}$ , but you wouldn't know it just by looking at it.

We saw that, if you have a rotation matrix, then, to determine the axis of rotation, you just need to solve $Mv=v$ for $v$ . We now introduce a generalisation of this equation.

Definition:

Suppose $M$ is an $n$ -by-$n$ matrix, $v\in\mathbf{C}^{n}$ is a complex vector and $\lambda\in\mathbf{C}$ is a complex number. If $Mv=\lambda v$ then we say that $v$ is an eigenvector of $M$ with eigenvalue $\lambda$ .

Remark:

We'll usually require $v\neq 0$ : this is because $v=0$ is always a solution to $Mv=\lambda v$ , so we don't really care about it (it tells us nothing interesting about $M$ ). We'll often abbreviate "eigenvector" to "evec" and "eigenvalue" to "eval". The prefix "eigen-" comes from German, and means "self": $v$ is sent to itself by $A$ , up to a scale factor $\lambda$ . You can use the prefix to construct many fun words, including eigenline (line of eigenvectors), eigendirection (direction in which eigenvector points), eigenspace (space consisting of eigenvectors for some particular eigenvalue), eigenfunctions (for example, $e^{\lambda x}$ is an eigenfunction of $d/dx$ because $d(e^{\lambda x})/dx=\lambda e^{\lambda x}$ .

It's hard to overemphasise the importance of eigenvalues and eigenvectors in mathematics and its applications. For example, in quantum mechanics, Schrödinger's equation is really just the statement that, if you have a quantum system, then the possible values that energy can take on are the eigenvalues of an operator called the Hamiltonian. This allows you predict the absorption/emission spectrum of hydrogen to great accuracy, which is one of the reasons people were so convinced by quantum mechanics early on. We'll focus on applications within mathematics, but this eigenstuff we're doing is really central in maths and in science.

The eigenvector equation

For each $\lambda$ , $Mv=\lambda v$ is an equation for $v$ : you fix an eigenvalue and you look for eigenvectors associated to that eigenvalue. You always find the solution $v=0$ , but very rarely do you find anything else. The $\lambda$ s for which $Mv=\lambda v$ has a nonzero solution are called the eigenvalues of $M$ . Not every number occurs as an eigenvalue of $M$ . In the next video we'll see how to figure out which numbers are eigenvalues for a given matrix.

In this video, we'll answer the question: if someone gives you an eigenvalue of $M$ , how do you find the eigenvectors?

Example:

If someone gives you $M=\begin{pmatrix}2&-1\\ 1&0\end{pmatrix}$ and tells you that $\lambda=1$ is an eigenvalue of $M$ , you can find the eigenvectors by solving $Mv=\lambda v=v$ : $Mv=\begin{pmatrix}2&-1\\ 1&0\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}$ $=\begin{pmatrix}2x-y\\ x\end{pmatrix},$ so $Mv=v$ implies $2x-y=x$ and $x=y$ . The second equation $x=y$ implies the first, so the eigenvectors are just the vectors of the form $\begin{pmatrix}x\\ x\end{pmatrix}$ .

All we did here was to solve a system of simultaneous equations for $v$ . So if someone gives you $\lambda$ , finding $v$ is something we know how to do. If we write this system in matrix form, we get $(M-\lambda I)v=0$ (i.e. the constants are all zero, not $\lambda v$ ).

What would happen if I picked something other than $\lambda=1$ ?

Example:

If I keep the same $M$ , pick $\lambda=2$ and try to solve $Mv=2v$ for $v$ , I get: $2x-y=2x,\qquad x=2y,$ which implies $y=0$ and $x=2y=0$ , so $v=0$ . This means that $2$ is not an eigenvalue of $M$ .

Currently we're just guessing $\lambda$ , and there are infinitely many possible $\lambda$ s, so we need a better way of figuring out which $\lambda$ s are eigenvalues of $M$ . That's what we'll do in the next video.