How to demonstrate the orthogonality of OFDM subcarriers?
Orthogonality is defined as &quot;the inner product of two vectors equals zero&quot;.
Now, in OFDM, the transmit vector for a single subcarrier is exactly one row vector $\mathbf D_k$ of the DFT Matrix $\mathbf D$, multiplied by the complex value of a symbol $c_k$, i.e. $c_K \mathbf D_K$.
Two different subcarriers $k, l, k\ne l$ hence have the inner product $\langle c_k\mathbf D_k,c_l\mathbf D_l\rangle$; inner products are linear things, hence that's
\begin{align}
\langle c_k\mathbf D_k,c_l\mathbf D_l\rangle &amp;= c_kc_l \langle \mathbf D_k,\mathbf D_l\rangle\\
&amp;= c_kc_l \begin{cases}0&amp;k\ne l\\\|\mathbf D_k\| &amp; k = l\end{cases} &amp;\text{q.e.d.},
\end{align}
because the DFT matrix is unitary.