On a Local Lipschitz Constant of the Maps Related to $LU$-Decomposition

Abstract

Let $M(n, \mathbb R)$ be the set of real positive definite symmetric $(n \times n)$-matrices equipped with the Euclidean norm, and let $A \in M(n, \mathbb R)$. Let $L(n, \mathbb R)$ be the set of all real non-degenerate lower-triangular $(n \times n)$-matrices equipped with the Euclidean norm, and let $L : M(n, \mathbb R) \to L(n, \mathbb R)$ be a (differentiable) map assigning to a positive definite symmetric matrix its lower-triangular factor in the $LU$-decomposition. We give an effective upper estimate for $\|L’(A)\|$.