I'm following the lecture notes by Timo Weigand on QFT.
On page 169, section 6.2 he is briefly touching on the non-abelian gauge symmetry in the SM.
The fundamental representation makes sense to me. For example, for $SU(3)$, we define the object or column vector with three component, suppressing spinor indices, $\psi(x)=(\psi_1, \psi_2, \psi_3)^T$. The fundamental representation, a $3*3$ matrix $V(h)$, acts on this column vector, with each component itself a Dirac spinor that has 4 components of complex numbers. Lagrangian density then is showed to remain invariant when the $\psi$ is multiplied by the $3*3$ matrix $V(h)$.
But the next step puzzles me a little, when we use the adjoint representation. Here instead of a 3 component column vector, we use a $3*3$ matrix $\psi(x)$, whose elements $\psi_{ij}(x)$are complex numbers based on the definition given in Eqn.(6.32).$$\forall x: \psi(x) \equiv \psi_{i j}(x) \in\left\{\mathbb{C}^{N, N} \mid \psi^{\dagger}(x)=\psi(x), \operatorname{tr} \psi(x)=0\right\}$$
It is a $3*3$ matrix or representation of the $SU(3)$ Lie algebra.
But when in Eqn.(6.34)$$\mathcal{L}=\operatorname{tr} \bar{\psi}(i \partial-m) \psi \equiv \operatorname{tr}\left[\psi_{i j}(i \partial-m) \psi_{j l}\right] \equiv \psi_{i j}(i \partial-m) \psi_{j i} $$the Lagranian density is defined as $tr(\bar{\psi}(i\gamma^\mu\partial_\mu-m)\psi)= \bar{ \psi_{ij}}(i\gamma^\mu\partial_\mu-m)\psi_{ji}$. Here $\psi$ is a $3*3$ matrix whose elements, $\psi_{ji}$, are not complext numbers but Dirac spinors.
In another word, is this what we mean by the spinor indices are suppressed in the adjoint representation? How can we think of $\psi(x)$ both as a $3*3$ matrix with components to be both complex numbers as defined in (6.32) and Dirac spinors as used in (6.34) that have themselves four components, each a complex number.
