# Spectral radius of sum of commuting matrices

• Jul 2nd 2010, 01:04 PM
chrysi
Spectral radius of sum of commuting matrices
If two matrices A and B commute, when is it true that rho(A+B) = rho(A)+rho(B), i.e. that the spectral radius of A+B is equal to the sum of the two spectral radii or A and B?
• Jul 2nd 2010, 01:18 PM
Bruno J.
What if $B=-A$?
• Jul 2nd 2010, 01:31 PM
chrysi
Thanks, with B = -A I guess it holds trivially, is there any other set of conditions that ensures this?
• Jul 2nd 2010, 01:33 PM
Bruno J.
Why does it hold? The spectral radius of the zero matrix is zero; so if your formula held, every matrix would have a zero spectral radius.
• Jul 2nd 2010, 01:38 PM
chrysi
shoot! of course

but is it true at least that if A and B commute then rho(A+B)<=rho(A)+rho(B)? I think yes. My question is really, when does this hold with equality.
• Jul 3rd 2010, 08:47 PM
Bruno J.
Quote:

Originally Posted by chrysi
shoot! of course

but is it true at least that if A and B commute then rho(A+B)<=rho(A)+rho(B)? I think yes. My question is really, when does this hold with equality.