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?

Printable View

- Jul 2nd 2010, 12:04 PMchrysiSpectral 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, 12:18 PMBruno J.
What if $\displaystyle B=-A$?

- Jul 2nd 2010, 12:31 PMchrysi
Thanks, with B = -A I guess it holds trivially, is there any other set of conditions that ensures this?

- Jul 2nd 2010, 12:33 PMBruno 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, 12:38 PMchrysi
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, 07:47 PMBruno J.