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 $\Vert L’(A)\Vert$.