How to Work with Eigenvectors and Eingenvalues - dummies

How to Work with Eigenvectors and Eingenvalues

By Steven Holzner

In quantum physics, when working with kets, it is useful to know how to use eigenvectors and eigenvalues. Applying an operator to a ket can result in a new ket:


To make things easier, you can work with eigenvectors and eigenvalues (eigen is German for “innate” or “natural”). For example,


is an eigenvector of the operator A if

  • The number a is a complex constant


Note what’s happening here: Applying A to one of its eigenvectors,


multiplied by that eigenvector’s eigenvalue, a.

Although a can be a complex constant, the eigenvalues of Hermitian operators are real numbers, and their eigenvectors are orthogonal


Casting a problem in terms of eigenvectors and eigenvalues can make life a lot easier because applying the operator to its eigenvectors merely gives you the same eigenvector back again, multiplied by its eigenvalue — there’s no pesky change of state, so you don’t have to deal with a different state vector.

Take a look at this idea, using the R operator from rolling the dice, which is expressed this way in matrix form:


The R operator works in 11-dimensional space and is Hermitian, so there’ll be 11 orthogonal eigenvectors and 11 corresponding eigenvalues.

Because R is a diagonal matrix, finding the eigenvectors is easy. You can take unit vectors in the 11 different directions as the eigenvectors. Here’s what the first eigenvector,


would look like:


And here’s what the second eigenvector,


would look like:


And so on, up to


Note that all the eigenvectors are orthogonal.

And the eigenvalues? They’re the numbers you get when you apply the R operator to an eigenvector. Because the eigenvectors are just unit vectors in all 11 dimensions, the eigenvalues are the numbers on the diagonal of the R matrix: 2, 3, 4, and so on, up to 12.