Lemma 1, p. 3 — for an admissible selection, P = 𝔼[P_𝔖] is positive definite with λmin(P) > 0 and λmax(P) ≤ 1

Let $n\ge1$ and let $\mathfrak S$ be a random subspace of $\mathbb R^n$ on a probability space $(\Omega,\mathcal A,\mathbb P)$, such that $\omega\mapsto P_{\mathfrak S(\omega)}x$ is measurable for each $x$. Suppose $\mathfrak S$ is admissible, i.e. $\mathbb P[x\in\mathfrak S^\perp]<1$ for every $x\neq0$, and let $\mathsf P=\mathbb E[P_{\mathfrak S}]$ be the average projection. Then $\mathsf P$ is a symmetric positive definite matrix, and its extreme eigenvalues satisfy

Precisely: $\mathsf P$ is self-adjoint, $\langle x,\mathsf Px\rangle>0$ for every $x\ne0$, and the Rayleigh extremes satisfy the two bounds above.

The average projection and its extreme eigenvalues govern the whole analysis of RPSD: $\lambda_{\min}(\mathsf P)>0$ is what makes the contraction factor of Theorem 1 strictly less than one, and positive definiteness is what makes $\mathsf Q=\mathsf P^{-1/2}$ well defined.

Formalization Note. The hypothesis $n\ge1$ is added: on $\mathbb R^0$ there are no eigenvalues, and the Rayleigh infimum over the empty unit sphere is $0$. "Positive definite matrix" is read as self-adjoint plus $\langle x,\mathsf Px\rangle>0$ for $x\neq0$. The measurability hypothesis guarantees that the Bochner integral defining $\mathsf P$ is the true expectation.