Linear Algebra and Its Applications · Problem 6.2.29

Positive Definite Matrices — Gilbert Strang

Mathematics · tests-for-positive-definiteness

Problem

The symmetric factorization A=LDLTA = LDL^T means that xTAx=xTLDLTxx^T A x = x^T LDL^T x: [xy][abbc][xy]=[xy][10b/a1][a00(ac−b2)/a][1b/a01][xy].\begin{bmatrix} x & y \end{bmatrix} \begin{bmatrix} a & b \\ b & c \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x & y \end{bmatrix} \begin{bmatrix} 1 & 0 \\ b/a & 1 \end{bmatrix} \begin{bmatrix} a & 0 \\ 0 & (ac-b^2)/a \end{bmatrix} \begin{bmatrix} 1 & b/a \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}. The left-hand side is ax2+2bxy+cy2ax^2 + 2bxy + cy^2. The right-hand side is a(x+bay)2+‾y2a\left(x + \frac{b}{a}y\right)^2 + \underline{\quad} y^2. The second pivot completes the square! Test with a=2a = 2, b=4b = 4, c=10c = 10.

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