Simultaneous Diagonalization of Hermitian Matrices
by MARYJANE D. MADULARA
In an earlier post about the properties of Hermitian operators, it was noted that quantum operators of physical significance are Hermitian by type. Here we discuss more fully about Hermitian matrices.
A n x n matrix is Hermitian if it is equal to its corresponding adjoint matrix. Now, for each Hermitian matrix, it may be diagonalized by a unitary transformation to its basis. That is by using a unitary matrix composed of eigenvectors of the Hermitian matrix.
But what can be done for two Hermitian matrices?
The good thing is that they may be simultaneously diagonalized. This can be done by finding the eigenvectors common to both. And then by verifying that under a unitary transformation to this basis, both matrices are diagonalized.
Let us consider the following Hermitian matrices.
EIGENVALUES AND EIGENVECTORS
i) For 
Look first for the eigenvalue 
by solving it from the determinant,
So that by using the basket rule in solving matrices,
This will give us the values, 
a. for 
For simplicity, first choose 
Next, choose 
b. for 
So choose 
,
ii) For 
Again look first for the eigenvalue 
by solving it from the determinant,
Then by using again the basket rule for matrices,
This will give us the values,
a. for 
The resulting equation will then be,
This results to,
Now choose 
so that,
b. for 
Then choose the values to be 
c. for 
The resulting equation will then be,
This will give us,
Since 
the second term will cancel out to zero, so that this will only then become,
Then choose 
so that,
So here are the common eigenvectors of 
and 
UNITARY TRANSFORMATION
Now for the Unitary transformation matrix,
Verify if 
Finally, using this unitary transformation, find out if 
and 
are diagonalized.
Thus we have described the properties of Hermitian operators in terms of its eigenvalues and eigenvectors. We have also shown that two Hermitian matrices can both be diagonalized through a unitary transformation.
About the Author:
Maryjane D. Madulara is presently pursuing a masters degree in physics at MSU-Iligan Institute of Technology (MSU-IIT) in Iligan City, Philippines. Computational physics research is her subject of interest. “Something new for the scientific community” is her motivation to continue, dream big, and do more. She hopes to finish a doctoral degree abroad.
August 5th, 2009 at 9:51 pm
the presentation is very nice..tnx diay sa pag-post te..though there are some few errors in the typing..but i think errors are sometimes necessary so the reader would know if he/she understands the contents..:d
August 5th, 2009 at 10:02 pm
Hello Ms. Mary Jane. I find your article very informative. I have really learned the basic steps on how to simultaneously diagonalize two Hermitian matrices. I only have a question though on what is your basis in selecting the values for the arbitrary variables? It seems that the Gram-Schmidt procedure for acquiring orthogonal vectors was not performed. How can I be sure that the values of the variables I selected are already orthogonal? thank you so much and more power.
August 5th, 2009 at 10:05 pm
The simultaneous diagonalization of hermitian matrices would have been perfect if there weren’t any mistakes on the matrix elements and eigenvalues. Although these were just clerical errors but these could cause confusion especially to those who are new to this subject. Thank You!
August 5th, 2009 at 10:11 pm
Good day to you ate MJ. Thanks to this article, it really cultivated me to love and appreciate hermitian operators because of its elegance and for being a special operator. Although there were minimal errors on the sign along the calculations but its acceptable for me.
Congratulations!
August 11th, 2009 at 12:17 am
The process on diagonalizing the two hermitian matrices was presented in a way that the reader could understood. The minimal errors encourage me to really study the topic well. Thanks for the effort.
August 11th, 2009 at 12:19 am
I don’t mean that you made an effort to make the minimal errors…hahaha…
what i mean is thank you for the effort of showing the process on simultaneous diagonalization. This will reslly help a lot in our xeam.
Thanks again:).
August 14th, 2009 at 7:43 am
Hi ate MJ! Thanks for that article. It’s very informative indeed! However, I noticed some clerical errors but anyway the important thing is that you presented it in a manner that newbies could understand.
December 10th, 2009 at 2:54 pm
Hello, MARYJANE. I just want to point out that the simultaneous denationalization works here because your matrices commute (\Omega \Gamma = \Gamma \Omega). In general, we can’t simultaneously diagonalize two matrices. Please refer to page 235 of the book ‘Matrix Analysis’ by Horn R A. And email me if you want to discuss more about this important research issue.
April 5th, 2010 at 4:53 pm
well im not too sure but i think you have tricked us here….
in practice one doesnt happen to choose the values for the e’vectors of the degenerate matrix that fits the other one…!
April 7th, 2010 at 5:12 pm
@shahab: I don’t see any trick on this. Everything is just fine. The eigenvectors were not just “chosen” but actually solved from the other matrix (\Lambda) which commutes with the matrix (\Omega) you wanted to diagonalize. The only important thing missing here is that it wasn’t mentioned that this method only works if both matrices commute.
April 27th, 2010 at 10:09 am
I see…
thanks.