Existence and Uniqueness of the Positive Semi-Definite Square Root

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 suppose that $V$ is \reftext{def:finite-dimensional-vector-space-2026b}{finite-dimensional} and $V\ne\{0_{V}\}$. Let $T$ 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}.

Then there is exactly one linear operator $R$ on $V$ that is self-adjoint, positive semi-definite, and satisfies

Since the product $RR$ of \ref{def:operator-operations-2026a} is the map sending $x\in V$ to $R(R(x))$, the displayed condition says exactly that $RR=T$.