Linear Algebra and Its Applications · Problem 6.2.19

Positive Definite Matrices — Gilbert Strang

Mathematics · tests-for-positive-definiteness

Problem

Which 3 by 3 symmetric matrices AA produce these functions f=xTAxf = x^T A x? Why is the first matrix positive definite but not the second one? (a) f=2(x12+x22+x32−x1x2−x2x3)f = 2(x_1^2 + x_2^2 + x_3^2 - x_1 x_2 - x_2 x_3). (b) f=2(x12+x22+x32−x1x2−x1x3−x2x3)f = 2(x_1^2 + x_2^2 + x_3^2 - x_1 x_2 - x_1 x_3 - x_2 x_3).

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