  We have investigated the excitation spectra of $j_{eff}$=$\frac{1}{2}$ Mott
insulator Na$_2$IrO$_3$. Taking into account a relativistic multiplet structure
of Ir ions, we have calculated the optical conductivity $\sigma(\omega)$ and
resonant inelastic x-ray scattering (RIXS) spectra, which manifest different
features from those of a canonical $j_{eff}$=$\frac{1}{2}$ system
Sr$_2$IrO$_4$.Distinctly from the two-peak structure in Sr$_2$IrO$_4$,
$\sigma(\omega)$ in Na$_2$IrO$_3$ has a broad single peak dominated by
interband transitions from $j_{eff}$=$\frac{3}{2}$ to $\frac{1}{2}$. RIXS
spectra exhibit the spin-orbit (SO) exciton that has a two-peak structure
arising from the crystal-field effect, and the magnon peak at energies much
lower than in Sr$_2$IrO$_4$. In addition, a small peak near the optical
absorption edge is found in RIXS spectra, originating from the coupling between
the electron-hole ($e$-$h$) excitation and the SO exciton. Our findings
corroborate the validity of the relativistic electronic structure and
importance of both itinerant and local features in Na$_2$IrO$_3$.
