Linear Algebra and Its Applications · Problem 6.2.22

Positive Definite Matrices — Gilbert Strang

Mathematics · tests-for-positive-definiteness

Problem

A diagonal entry

a_{jj}

of a symmetric matrix cannot be smaller than all

\lambda

's. If it were, then

A - a_{jj} I

would have eigenvalues and would be positive definite. But

A - a_{jj} I

has a on the main diagonal.