Is the attenuation constant in EM wave equation a vector or tensor?

In general, an EM wave within a lossy medium is expressed as
$mathbf{E}=mathbf{E_0}e^{-gamma_x x-gamma_y y-gamma_z z}$.
If we consider $mathbf{gamma}=gamma_x hat{x}+gamma_y hat{y}+gamma_z hat{z}$ and $mathbf{r}=xhat{x}+yhat{y}+zhat{z}$, the above equation is expressed as
$mathbf{E}=mathbf{E_0}e^{-gammacdotmathbf{r}}=mathbf{E_0}e^{-(alpha+jbeta)cdotmathbf{r}}$, because $gamma=alpha+jbeta$. $beta$ is also considered as a vector and due to the dot product between the $beta$ and $mathbf{r}$ the phase propagation of the wave takes place in three-dimensional space. As the dot product is operating over the $(alpha+jbeta)$, an $alphacdot mathbf{r}$ should also be expected. But for that, the $alpha$ must be a vector. But, $alpha$ contains terms associated with the permittivity and permeability which are tensors. Therefore, I am a bit confused about whether the $alpha$ also be a vector or tensor?

Please help me to understand the right thing.