Given an $M\times L (M>L)$ matrix $$\mathbf{A}=(\mathbf{a}_1,\mathbf{a}_2,\cdots,\mathbf{a}_L)$$ where
$$\mathbf{a}_k = (1,e^{j2\pi c\sin\theta_k},e^{j2\pi2c\sin\theta_k},\cdots,e^{j2\pi(M-1)c\sin\theta_k})$$
is an $M \times 1$ complex vector of length $\sqrt{M}$, $j=\sqrt{-1}$, $\theta_k\in [-\pi/2,\pi/2]$, $c$ is a real constant and the value of the $L$ angles $\theta_k (k=0,1,\cdots,L-1)$ are all different, let $$\mathbf{a}_p = \mathbf{A}(\mathbf{A}^H\mathbf{A})^{-1}\mathbf{A}^H\mathbf{a}$$ be the projection of a column vector $\mathbf{a}$, which has the same form with $\mathbf{a}_k$ but with a different angle $\theta$, onto the column space of $\mathbf{A}$, I want to find the relationship between the norm of the projection
$$\frac{||\mathbf{a}_p||^2_2}{M}$$
and the vector dimension $M$. Numerical tests show that the above expression decreases as $M$ increases. So I guess it also holds analytically, but I don't know how to prove it?
Can someone help. Thanks.
P.S.:
- Can the above result generalize to any complex vector $\mathbf{a}\in C^{M\times 1}$ with $||\mathbf{a}||=\sqrt{M}$?
- This question is closely related to the post I ask.