A recent post reminded me of a several year old question I have. I think I know the answer, but I wanted your thoughts on it.

I was doing a homework assignment in Quantum Physics and one of the problems was to diagonalize a unitary Hermitian matrix. When I did this he claimed that the order of the elements in the diagonal was incorrect.

Now, I typically don't care about the order of the elements in the diagonal. In Quantum we don't care about the order because the difference in order is represented by a similarity transformation between the two matrices, which has no effect on the Physics. When I pointed this out to my professor he gave me half the points back for the "error."

However I never did check to see if there is some established paradigm for ordering the diagonal.

So my question is this:

When you diagonalize the matrix:

$\displaystyle \left ( \begin{array}{cc} -3 & 0 \\ 2 & -1 \end{array} \right )$

should the answer be

$\displaystyle \left ( \begin{array}{cc} -1 & 0 \\ 0 & -3 \end{array} \right )$

or

$\displaystyle \left ( \begin{array}{cc} -3 & 0 \\ 0 & -1 \end{array} \right )$

or is there no practical difference between the two?

-Dan