  This paper provides a classification theorem for expansive matrices $A \in
\mathrm{GL}(d, \mathbb{R})$ generating the same anisotropic homogeneous
Triebel-Lizorkin space $\dot{\mathbf{F}}^{\alpha}_{p, q}(A)$ for $\alpha \in
\mathbb{R}$ and $p,q \in (0,\infty]$. It is shown that
$\dot{\mathbf{F}}^{\alpha}_{p, q}(A) = \dot{\mathbf{F}}^{\alpha}_{p, q}(B)$ if
and only if the homogeneous quasi-norms $\rho_A, \rho_B$ associated to the
matrices $A, B$ are equivalent, except for the case $\dot{\mathbf{F}}^0_{p, 2}
= L^p$ with $p \in (1,\infty)$. The obtained results complement and extend the
classification of anisotropic Hardy spaces $H^p(A) =
\dot{\mathbf{F}}^{0}_{p,2}(A)$, $p \in (0,1]$, in [Mem. Am. Math. Soc. 781, 122
p. (2003)].
