On the positive definite solutions of the matrix equations \(X^{s}\pm A^{\text T} X^{-t} A=I_{n}\)
The paper deals with the matrix equations \(X^s+\sigma A^\top X^{-t}A=I_n\), where \(\sigma=\pm 1\), \(s,t\) are positive integers, \(I_n\) is the identity \(n\times n\) matrix and \(X\in \mathbb{R}^{n\times n}\) is the solution. The case \(s=1\) is well studied in the literature. Using the Brouwer fixed point principle the authors prove theorems for the existence of symmetric positive definite solutions. Condition numbers for the solutions of these equations are derived and other perturbation results are also obtained. Finally the authors consider the iterative schemes \(X_{k+1}=(I_n- \sigma A^\top X_k^{-t}A)^{1/s}\) for computing the solutions.
scientific article; zbMATH DE number 1956793