This exercise is concerned with a possible action of the additive group of real numbers on the space of positive definite matrices of order \(n\). Let \(\mathcal{X} \subset \mathbb{R}^{n}\) be a given subspace. To each \(\Sigma\) and \(W=\Sigma^{-1}\) there corresponds a \(W\)-orthogonal projection \(P_{W}\) whose image is \(\mathcal{X}\), and a complementary projection \(Q_{W}=I-P_{\mathcal{X}}\). In matrix notation, \(P_{W}=X\left(X^{\prime} W Xight)^{-1} X^{\prime} W\) depends on \(\Sigma\). For \(g \in \mathbb{R}\), show that the transformations

\[
\Sigma \stackrel{g}{\longmapsto} Q_{W} \Sigma+e^{g} P_{W} \Sigma=\Sigma+\left(e^{g}-1ight) X\left(X^{\prime} \Sigma^{-1} Xight)^{-1} X^{\prime}
\]

determine a group homomorphism by linear transformations on the space of positive-definite matrices.