Matrix extremal problems and shift invariant means

We consider an m×m real symmetric matrix M_a(x) with a₁,…,a_m on the main diagonal and x in all off-diagonal positions, where m≥2 and a=(a₁,…,a_m) is a given m-tuple of positive real numbers. We study the extremal problem of finding the minimum and maximum of x where M_a(x) is positive semidefinite. We show that the polynomial det⁡M_a(x) in variable x has only real roots with a unique negative root and that M_a(x) is positive semidefinite (resp. definite) if and only if x lies in the closed (resp. open) interval determined by the negative and smallest positive roots. It is further shown that the negative and smallest positive root maps over m-tuples of positive real numbers contract the Thompson metric and induce new multivariate means of positive real numbers satisfying the monotonicity, homogeneity, joint concavity and super-multiplicativity. In particular, the smallest positive root map extends to such a mean of infinite variable of positive real numbers that realizes the limits for decreasing sequences, and it eventually gives rise to a shift invariant mean of bounded sequences.