Linear Algebra and Its Applications · Problem 6.2.43

Positive Definite Matrices — Gilbert Strang

Mathematics · tests-for-positive-definiteness

Problem

A group of nonsingular matrices includes

AB

and

A^{-1}

if it includes

A

and

B

. ``Products and inverses stay in the group.'' Which of these sets are groups? Positive definite symmetric matrices

A

, orthogonal matrices

Q

, all exponentials

e^{tA}

of a fixed matrix

A

, matrices

P

with positive eigenvalues, matrices

D

with determinant 1. Invent a group containing only positive definite matrices.