On the existence of a positive definite solution of the matrix equation \(X+A^ T X^{-1} A=I\)
First, the author demonstrates that the problem described in the title is equivalent to the recursion problem \(\forall n \in \mathbb{N}\), is \(X_ n > AA^ T\) if \(X_{n+1} = I -A^ T X_ n^{-1}A\) with \(X_ 0= I\). From this follow not only necessary solvability conditions in terms of the spectral radius of \(A\), \(A+A^ T\) and \(A-A^ T\), respectively, but also an algorithm for the numerical computation of the solution. For normal matrices \(A\) one of these conditions is necessary and sufficient. Moreover, it is demonstrated that the matrix \(A\) need not be regular and how this equation can be used to solve an indefinite linear/quadratic, discrete-time optimal control problem. The paper is well-organized, and easy to read.
indefinite linear/quadratic, discrete-time optimal control problem
scientific article; zbMATH DE number 540059