Linear Algebra and Its Applications · Problem 6.2.28

Positive Definite Matrices — Gilbert Strang

Mathematics · tests-for-positive-definiteness

Problem

In the Cholesky factorization

A = CC^T

, with

C = L\sqrt{D}

, the square roots of the pivots are on the diagonal of

C

. Find

C

(lower triangular) for

A = \begin{bmatrix} 9 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 2 & 8 \end{bmatrix} \quad \text{and} \quad A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 7 \end{bmatrix}.