# The smallest real part of eigenvales of weighted sum of two matrices

I have a matrix problem that I need help with.

Let H=aA+bB, where a+b=1,a>0,b>0,and A,B are matrices having non-negative real part eigenvalues. In addition, A+B has positive real part eigenvalues. I also know that all the eigenvalues of H have positive real parts. I want to get a lower bound on the minimum real part of the eigenvalues of H in terms of a,b and real parts of the eigenvalues of A+B.

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Without further information on how you know the eigenvalues are positive, I doubt that much can be said. In general, positivity of eigenvalues of aA+bB does not follow from positiivity of eigenvalues of A and B. –  Michael Renardy May 8 '12 at 13:16
I imagine there could be such a bound, which in addition would depend on the matrix size. –  Dima Pasechnik May 8 '12 at 22:24