Necessary and sufficient conditions for the existence of positive definite solutions of the matrix equation \(X+A^Tx^{-2}A=I\). (English) Zbl 1083.15020
The author shows that the matrix equation as specified in the title has a symmetric and positive definite solution \(X\) if and only if the factorization \(A = W^TWZ\) may be performed, where \(W\) is a square non-singular matrix and the columns of \([W^T Z^T]^T\) are orthonormal. On this basis some properties of \(A\) as well as relations between \(X\) and \(A\) are derived. The results may be useful for solving matrix equation arising from finite-difference approximation to an elliptic differential equation.
Reviewer: Mihail Voicu (Iaşi)
MSC:
| 15A24 | Matrix equations and identities |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
Keywords:
matrix equation; matrix decomposition; positive definite solution; finite-difference approximation; elliptic differential equationReferences:
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