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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:

image0.png

First, you can rewrite this equation as the following:

image1.png

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

image2.png

Remember that the solution to

image3.png

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:

image4.png

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

image5.png

Next, find the determinant:

image6.png

And this can be factored as follows:

image7.png

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:

image8.png

So you have

image9.png

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

image10.png

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

image11.png

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

image12.png

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

image13.png

Then you have

image14.png

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

image15.png

Drop the arbitrary constant:

image16.png

So the eigenvalues of this matrix operator

image17.png

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

image18.png

The eigenvector corresponding to a2 is

image19.png
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