An Operator with an Orthonormal Eigenbasis is Positive Semi-Definite Exactly When its Eigenvalues are Nonnegative

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$. Let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$, and let $e\in V^{n}$ be an $n$-tuple in $V$ that is an orthonormal basis of $V$, with components $e_{k}$. Let $\lambda\in\mathbb{C}^{n}$ be an $n$-tuple with components $\lambda_{k}$ in the field $\mathbb{C}$ of complex numbers, and let $T$ be a linear operator on $V$ satisfying

$$T(e_{k})=\lambda_{k}e_{k}\qquad\text{for every }k\in[n].$$

Such $e$ and $\lambda$ exist whenever $V$ is finite-dimensional with $V\ne\{0_{V}\}$ and $T$ is self-adjoint, by Spectral Theorem for a Self-Adjoint Operator in Finite Dimensions.

Then $T$ is positive semi-definite if and only if for every $k\in[n]$ the number $\lambda_{k}$ is a real number satisfying

$$0\le\lambda_{k},$$

the order being that of the ordered field of real numbers.