Linear Algebra and Its Applications · Problem 6.2.14

Positive Definite Matrices — Gilbert Strang

Mathematics · tests-for-positive-definiteness

Problem

Decide whether the following matrices are positive definite, negative definite, semidefinite, or indefinite: A=[123254349],B=[120026−200−25−200−23],C=−B,D=A−1.A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 5 & 4 \\ 3 & 4 & 9 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 2 & 0 & 0 \\ 2 & 6 & -2 & 0 \\ 0 & -2 & 5 & -2 \\ 0 & 0 & -2 & 3 \end{bmatrix}, \quad C = -B, \quad D = A^{-1}. Is there a real solution to −x2−5y2−9z2−4xy−6xz−8yz=1-x^2 - 5y^2 - 9z^2 - 4xy - 6xz - 8yz = 1?

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