Positive Definite Operator

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-inner-product-space-2026a}{complex inner product space} with \reftext{lem:vector-space-basic-identities-2026a}{zero vector} $0_{V}$, and let $T$ be a \reftext{def:linear-operator-2026a}{linear operator} on $V$.

The operator $T$ is \textbf{positive definite} if for every $u\in V$ the complex number $\langle u,T(u)\rangle$ is a \reftext{def:real-numbers-c54-2026c}{real number} satisfying

the order being that of the \reftext{def:ordered-field-c54-2026b}{ordered field} of real numbers, and if moreover $\langle u,T(u)\rangle=0$ holds only for $u=0_{V}$.