On a local Lipschitz constant of the maps related to \(LU\)-decomposition (Q5938144)

Assume \(M(n,\mathbb{R})\) is the set of real positive definite symmetric \((n\times n)\)-matrices equipped with the Euclidean norm, and let \(A\in M(n,\mathbb{R})\). Denote by \(L(n,\mathbb{R})\) 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.NEWLINENEWLINENEWLINEThe authors obtain an effective upper estimate for \(\|L'(A)\|\). The analysis follows some very beautiful inequalities which are derived in a gentle manner.