Let ${\bf A} \in \mathbb{C}^{M\times N}$ be a Vandermonde matrix
\begin{equation}
\bf A = \begin{bmatrix}1&1&\cdots&1 \\
z_1&z_2&\cdots&z_N\\
\vdots&\vdots&\ddots&\vdots\\
z_1^{M-1}&z_2^{M-1}&\cdots&z_N^{M-1}
\end{bmatrix}
\end{equation}
where $z_n=e^{i\omega_n}$.
It is known that the rank of $\bf A $ is $N$ if $M\geq N$ and $z_m\neq z_n$ when $m\neq n$. Is there any formal proof?
Thanks.
Best Answer
It suffices to show that the first $N$ rows are linearly independent. In this regard, you may see ProofWiki for two different proofs.