Linear Algebra Done Right · Problem 7C.4

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · positive-operators

Problem

Suppose

n

is an integer with

n > 1

. Show that there exists an

n

-by-

n

matrix

A

such that all of the entries of

A

are positive numbers and

A = A^*

, but the operator on

\mathbf{F}^n

whose matrix (with respect to the standard basis) equals

A

is not a positive operator.