Consider an equilibrium state of a three-dimensional linear system, namely assume the case where the three eigenvalues $\lambda_i, i=1,2,3$, are real and $\lambda_3 < \lambda_2 <\lambda_1<0$. Then, the assocciated three-dimensional system may be reduced to the form $$ \dot{x}=\lambda_1 x,~~~\dot{y}=\lambda_2 y,~~~\dot{z}=\lambda_3 z. $$ Its general solution is given by $$ x=e^{\lambda_1 t}x_0,~~~y=e^{\lambda_2 t}y_0,~~~z=e^{\lambda_3 t}z_0. $$
Since all $\lambda_i$'s are negative, all trajectories tend to the origin as $t\to +\infty$. Furthermore, all trajectories outside of the non-leading plane $(y,z)$ approach the origin along the leading direction that coincides with the $x$-axis.
The following picture shows the phase portrait: The fewer the arrows the weaker is the rate of contraction. The leading subspace $E^L$ is one-dimensional, the two-dimensional subspace $E^{ss}$ is non-leading.
Could you please tell me why we call the $(y,z)$-plan non-leading and the $x$-axis leading? I cannot see it.