Suppose that we have vector $y_t = (z_t, x_t)'$ for $t = 1, 2, \dots, T$. Let $\Omega_t$ be the information set available at time $t$ and $z_t(h|\Omega_t)$ be optimal $h$-step predictor. $\Sigma_z(h|\Omega_t)$ denotes forecast MSE matrix.

We say that $x_t$ helps to predict $z_t$ (Granger Causality) if $$\Sigma_z\left(h|\Omega_t\setminus\{x_s|s\leq t\}\right) - \Sigma_z(h|\Omega_t)$$ for at least one $h = 1,2, \dots$ is positive semidefinite and $$\Sigma_z(h|\Omega_t) \neq \Sigma_z\left(h|\Omega_t\setminus\{x_s|s\leq t\}\right).$$ The definition is clear for me. $x_t$ helps predicting $z_t$ if the covariance matrix of the prediction without using $x_t$ is bigger then when we use $x_t$ in a matrix sense.

My confusion is more about interpretation of the semidefinite matrix. By definition $m\times m$ matrix $A$ is positive semidefinite iff for any $m\times 1$ vector $x$ $$x'Ax \geq 0.$$

Let $A \equiv \Sigma_z\left(h|\Omega_t\setminus\{x_s|s\leq t\}\right) - \Sigma_z(h|\Omega_t).$ So what is the intuition that if $x'Ax \geq 0$, then I can conclude that $x_t$ helps in predicting $z_t$?