A Positive Semi-Definite Square Root Acts on Eigenvectors by the Nonnegative Square Root

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-inner-product-space-2026a}{complex inner product space}, and let $R$ be a \reftext{def:linear-operator-2026a}{linear operator} on $V$ that is \reftext{def:self-adjoint-operator-2026b}{self-adjoint} and \reftext{def:positive-semidefinite-operator-2026a}{positive semi-definite}.

Let $\lambda$ be a \reftext{def:real-numbers-c54-2026c}{real number} with $0\le\lambda$, the order being that of the \reftext{def:ordered-field-c54-2026b}{ordered field} of real numbers, and let $\mu$ be the unique real number with $0\le\mu$ and $\mu^{2}=\lambda$, which exists and is unique by \ref{thm:real-nonnegative-square-root-2026a}; here $\mu^{2}$ abbreviates the product $\mu\cdot\mu$ in the \reftext{def:field-c54-2026b}{field} of real numbers. Real numbers are \reftext{def:complex-numbers-2026a}{complex numbers} by condition 1 of \ref{def:complex-numbers-2026a}, so $\lambda v$ and $\mu v$ are defined for $v\in V$.

Let $v\in V$ satisfy

Then