zeros of sum of two (matrix-valued!) polynomials
I have a nontrivial problem. let f(z) and g(z) be two (possibly monic) matrix-valued polynomials (i.e. of the form with being matrices, and ). assume that det(f(z)) and det(g(z)) has all real and negative roots. assume det(f(z)+g(z)) has all roots real.
prove (or disprove) that det(f(z)+g(z)) cannot have a positive zero.
for the case of dimension 1, it's easy since f(z) and g(z) are positive for z>0, and so f(z)+g(z) is positive for z>0, so all the roots are negative.
any suggestion/idea/reference would be highly appreciated...