Linear Algebra and Its Applications · Problem 7.2.8

Computations with Matrices — Gilbert Strang

Mathematics · cholesky-condition-number

Problem

For a positive definite

A

, the Cholesky decomposition is

A = LDL^T = R^TR

, where

R = \sqrt{D}L^T

. Show directly from equation (12) that the condition number of

c(R)

is the square root of

c(A)

. Elimination without row exchanges cannot hurt a positive definite matrix, since

c(A) = c(R^T)c(R)