On descartes’ rule of signs for matrix polynomials

We present a generalized Descartes’ rule of signs for self-adjoint matrix polynomials whose coefficients are either positive or negative definite, or null. In particular, we conjecture that the number of real positive (negative) eigenvalues of a matrix polynomial is bounded above by the product of the size of the matrix coefficients and the number of definite sign alternations (permanences) between consecutive coefficients. Our main result shows that this generalization holds under the additional assumption that the matrix polynomial is hyperbolic. In addition, we prove individual cases where the matrix polynomial is diagonalizable by congruence, or of degree three or less. The full proof of our conjecture is an open problem; we discuss analytic and algebraic approaches for solving this problem and ultimately, what makes this open problem non-trivial. Finally, we prove generalizations of two famous extensions of Descartes’ rule: If all eigenvalues are real then the bounds in Descartes’ rule are sharp and the number of real positive and negative eigenvalues have the same parity as the associated bounds in Descartes’ rule.