Linear Algebra and Its Applications · Problem 6.2.12

Positive Definite Matrices — Gilbert Strang

Mathematics · tests-for-positive-definiteness

Problem

In three dimensions,

\lambda_1 y_1^2 + \lambda_2 y_2^2 + \lambda_3 y_3^2 = 1

represents an ellipsoid when all

\lambda_i > 0

. Describe all the different kinds of surfaces that appear in the positive semidefinite case when one or more of the eigenvalues is zero.