How to Find the Eigenvectors and Eigenvalues of an Operator

In quantum physics, if you’re given an operator in matrix form, you can find its eigenvectors and eigenvalues. For example, say you need to solve the following equation:

First, you can rewrite this equation as the following:

I represents the identity matrix, with 1s along its diagonal and 0s otherwise:

Remember that the solution to

exists only if the determinant of the matrix A – aI is 0:

det(A – aI) = 0

How to find the eigenvalues

Any values of a that satisfy the equation det(A – aI) = 0 are eigenvalues of the original equation. Try to find the eigenvalues and eigenvectors of the following matrix:

First, convert the matrix into the form A – aI:

Next, find the determinant:

And this can be factored as follows:

You know that det(A – aI) = 0, so the eigenvalues of A are the roots of this equation; namely, a₁ = –2 and a₂ = –3.

How to find the eigenvectors

How about finding the eigenvectors? To find the eigenvector corresponding to a₁, substitute a₁ — the first eigenvalue, –2 — into the matrix in the form A – aI:

So you have

Because every row of this matrix equation must be true, you know that

And that means that, up to an arbitrary constant, the eigenvector corresponding to a₁ is the following:

Drop the arbitrary constant, and just write this as a matrix:

How about the eigenvector corresponding to a₂? Plugging a₂, –3, into the matrix in A –aI form, you get the following:

Then you have

And that means that, up to an arbitrary constant, the eigenvector corresponding to a₂ is

Drop the arbitrary constant:

So the eigenvalues of this matrix operator

are a₁ = –2 and a₂ = –3. And the eigenvector corresponding to a₁ is

The eigenvector corresponding to a₂ is