lou pecora

This is a math question, but it has ****ogs in Quantum Mechanics.

Given a matrix A (symmetric in my case), are their standard (optimal?)
approaches to finding the set of all (symmetric) matrices that commute
with A, i.e. {C | [A,C]=0} ? Is it as simple as getting the matrix S
that diagonalizes A and then "transforming back" all possible diagonal
matrices with real numbers on the diagonal? I.e., C=SDS^-1 for all
diagonal D? Or are there some other, better approaches?

Just point me in the right direction if you have any info.

Thanks in advance.

-- Lou Pecora (my views are my own) REMOVE THIS to email me.

kloeckner_benoît

No, this is not exact. But the idea is good. Up to this change of
basis, we can suppose that A is a diagonal matrix. Then a simple
computation shows that if A has the form:

/ l_1 \ with blocks of dimension
| l_1 | m_1,...,m_n
| ... |
| l_1 |
| l_2 |
| ... |
| l_2 |
| ... |
\ l_n /

then the commutator of A is the set of all matrices that are of the form

/ B_1 \ where the A_i's are square blocks of dimension
| B_2 | m_n
| ... |
\ B_n /

Besides this (taking into account the multiplicity of the eigenvalues
of A), it is as simple as what you proposed.