The Matrix Golden Mean and Its Applications to Riccati Matrix Equations

In this paper we generalize the concept of the golden mean of positive numbers to the golden mean of positive definite matrices and apply it to some Riccati algebraic and differential matrix equations. We describe the unique positive definite solutions of the Riccati matrix equations XA⁻¹X ± X-(B-A)=0 with 0<A≤ B in terms of geometric and golden means of positive definite matrices as well as the asymptotic behavior of the associated Riccati differential equations X=-XA⁻¹X ∓ X+(B-A). We describe (apparently new) results related to matrix continued fractions, symplectic Hamiltonian matrices, and other matrix means obtained from golden mean–related inequalities via canonical iterative processes.