Linear Algebra and Its Applications · Problem 6.2.27

Positive Definite Matrices — Gilbert Strang

Mathematics · tests-for-positive-definiteness

Problem

With positive pivots in DD, the factorization A=LDLTA = LDL^T becomes LDDLTL\sqrt{D}\sqrt{D}L^T. (Square roots of the pivots give D=DDD = \sqrt{D}\sqrt{D}.) Then C=LDC = L\sqrt{D} yields the Cholesky factorization A=CCTA = CC^T, which is ``symmetrized LULU'': From C=[3012]C = \begin{bmatrix} 3 & 0 \\ 1 & 2 \end{bmatrix} find AA. From A=[48825]A = \begin{bmatrix} 4 & 8 \\ 8 & 25 \end{bmatrix} find CC.

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