Linear Algebra and Its Applications · Problem 6.2.31

Positive Definite Matrices — Gilbert Strang

Mathematics · tests-for-positive-definiteness

Problem

For the semidefinite matrices A=[2−1−1−12−1−1−12] (rank 2)andB=[111111111] (rank 1),A = \begin{bmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{bmatrix} \text{ (rank 2)} \quad \text{and} \quad B = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix} \text{ (rank 1)}, write xTAxx^T A x as a sum of two squares and xTBxx^T B x as one square.

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