= Solution
Let $H$ replicate $\xi_1$, so $\xi_1=H^TP_1$. The positive-definite <Gram matrix> $Q=\mathbb E[P_1P_1^T]$ is invertible. Multiplication by $P_1$ and taking <expected values> give
$$
\mathbb E[P_1\xi_1]=QH,\qquad
H=Q^{-1}\mathbb E[P_1\xi_1].
$$
All quantities are integrable: the finite-atom representation from completeness gives finite terminal values, and the matrix $Q$ has finite entries. Positive definiteness also guarantees uniqueness of holdings, since $h^TP_1=0$ would imply $h^TQh=0$ and hence $h=0$.
Thus the replication cost is
$$
H^TP_0=P_0^TQ^{-1}\mathbb E[P_1\xi_1]
=\mathbb E[(P_0^TQ^{-1}P_1)\xi_1].
$$
Consequently the <one-period Gram-matrix replication formula> is
$$
\boxed{\xi_0=\mathbb E[Z\xi_1],\qquad Z=P_0^TQ^{-1}P_1,\qquad W=Q^{-1}P_1,\quad H=\mathbb E[W\xi_1].}
$$
The symmetry of $Q^{-1}$ makes the displayed orientations consistent. Positivity of this $Z$ would require an additional no-arbitrage condition; positive definiteness of the payoff <Gram matrix> alone proves the representation and uniqueness, not positivity of prices across states.
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