Thread: 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?

What if ?

Thanks, with B = -A I guess it holds trivially, is there any other set of conditions that ensures this?

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.

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.

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.

Sorry for reading your question too fast.
It might be true!