Linear Algebra and Its Applications · Problem 6.2.21

Positive Definite Matrices — Gilbert Strang

Mathematics · tests-for-positive-definiteness

Problem

A positive definite matrix cannot have a zero (or even worse, a negative number) on its diagonal. Show that this matrix fails to have xTAx>0x^T A x > 0: [x1x2x3][411102125][x1x2x3] is not positive when (x1,x2,x3)=(  ,  ,  ).\begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix} \begin{bmatrix} 4 & 1 & 1 \\ 1 & 0 & 2 \\ 1 & 2 & 5 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \text{ is not positive when } (x_1, x_2, x_3) = (\;,\;,\;).

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