Existence and Uniqueness of the Positive Semi-Definite Square Root

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$, and suppose that $V$ is finite-dimensional and $V\ne\{0_{V}\}$. Let $T$ be a linear operator on $V$ that is self-adjoint and positive semi-definite.

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

$$R(R(x))=T(x)\qquad\text{for every }x\in V.$$

Since the product $RR$ of Operations on Linear Operators is the map sending $x\in V$ to $R(R(x))$, the displayed condition says exactly that $RR=T$.