Eigenvalues of 4x4 Hermitian Matrix (Observable)

[Gunthi]

Homework Statement

Find the allowed energies for a spin-3/2 particle with the given Hamiltonian:
$$\hat{H}=\frac{\epsilon_0}{\hbar}(\hat{S_x^2}-\hat{S_y^2})-\frac{\epsilon_0}{\hbar}\hat{S_z}$$

The Attempt at a Solution

The final matrix I get is:

\begin{pmatrix}
\frac{3}{2} & 0 & \hbar\sqrt{3} & 0\\
0& \frac{\hbar}{2}-\frac{1}{2} & 0 &\hbar\sqrt{3} \\
\hbar\sqrt{3}& 0 & \frac{\hbar}{2}+\frac{1}{2} & 0\\
0& \hbar\sqrt{3} & 0 & \frac{3}{2}
\end{pmatrix}

My question is: Is there a more quick way to find the eigenvalues of a 4x4 hermitian matrix than going trough the tedious calculation of $det(\hat{H}-\lambda I)=0$?

 

[tiny-tim]

Hi Gunthi!

Is there a more quick way to find the eigenvalues of a 4x4 hermitian matrix than going trough the tedious calculation of $det(\hat{H}-\lambda I)=0$?

If you swap the second and third rows, it becomes two 2x2 matrices.

 

[Gunthi]

Hi Gunthi!


If you swap the second and third rows, it becomes two 2x2 matrices.

Hi tiny-tim!

I would also have to switch the 2nd and 3rd columns right? Then I would just calculate the eigenvalues of the 2x2 matrices separately?

I've been searching for properties of block matrices that could justify this, but to no avail. Is there a theorem that demonstrates this property? Or could you explain how this works?

I guess I'm rustier than I thought at my algebra

 

[tiny-tim]

Hi Gunthi!

… I've been searching for properties of block matrices that could justify this, but to no avail. Is there a theorem that demonstrates this property? Or could you explain how this works?

We're only re-arranging …

instead of the basis x y z t (or whatever), we're using x z y t ​

To put it another way, can't you immediately see, just by looking at it, that the matrix is in two parts that operate completely separately?