For an $M\times M$ matrix
$$\mathbf{R}=\mathbf{I}+\sum_{k=1}^{L}\lambda_k^2\mathbf{a}_k\mathbf{a}_k^H=\mathbf{I}+\mathbf{A}\mathbf{\Lambda}\mathbf{A}^H$$
where $\mathbf{I}$ is an identity matrix, $\mathbf{\Lambda}$ is a diagonal matrix with real elements $\lambda_k^2\ge1$ and $\mathbf{A}=(\mathbf{a}_1,\mathbf{a}_2,\cdots,\mathbf{a}_L)$ with $\mathbf{a}_k$ being an $M\times1$ column vector whose norm is $\sqrt{M}$.
In addition, the $m$-th ($1\le m\le M$) element of $\mathbf{a}_k$ is given by $e^{j2\pi(m-1)c\sin\theta_k}$, where $j=\sqrt{-1}$, $\theta_k\in [-\pi/2,\pi/2]$ is an angle and $c$ is a real constant. The value of the $L$ angles are all different, thus $A$ is full column rank.
Let $\mathbf{a}_p$ be the projection of column vector $\mathbf{a}$, which has the same form with $\mathbf{a}_k$ but with a different angle $\theta$, onto the column space of $\mathbf{A}=(\mathbf{a}_1,\mathbf{a}_2,\cdots,\mathbf{a}_L)$.
Given the above background, I want to bound the following ratio
$$r = \dfrac{\mathbf{a}_p^H\mathbf{R}^{-1}\mathbf{R}^{-1}\mathbf{a}_p-\left(\mathbf{a}_p^H\mathbf{R}^{-1}\mathbf{a}_p/||\mathbf{a}_p||\right)^2}{M-||\mathbf{a}_p||^2+\mathbf{a}_p^H\mathbf{R}^{-1}\mathbf{R}^{-1}\mathbf{a}_p}$$
My first attempt:
$$r = \dfrac{\mathbf{a}_p^H\mathbf{R}^{-1}\mathbf{R}^{-1}\mathbf{a}_p-\left(\mathbf{a}_p^H\mathbf{R}^{-1}\mathbf{a}_p/||\mathbf{a}_p||\right)^2}{M-||\mathbf{a}_p||^2+\mathbf{a}_p^H\mathbf{R}^{-1}\mathbf{R}^{-1}\mathbf{a}_p}\le \dfrac{\mathbf{a}_p^H\mathbf{R}^{-1}\mathbf{R}^{-1}\mathbf{a}_p||\mathbf{a}_p||^2}{(M-||\mathbf{a}_p||^2)||\mathbf{a}_p||^2+\mathbf{a}_p^H\mathbf{R}^{-1}\mathbf{R}^{-1}\mathbf{a}_p||\mathbf{a}_p||^2}$$
Using the eigenvalue decomposition of $\mathbf{R}$, it is easy to verify
$$||\mathbf{a}_p||^2 \ge \mathbf{a}_p^H\mathbf{R}^{-1}\mathbf{R}^{-1}\mathbf{a}_p$$
Thus,
$$r \le \dfrac{\mathbf{a}_p^H\mathbf{R}^{-1}\mathbf{R}^{-1}\mathbf{a}_p||\mathbf{a}_p||^2}{(M-||\mathbf{a}_p||^2)||\mathbf{a}_p||^2+\mathbf{a}_p^H\mathbf{R}^{-1}\mathbf{R}^{-1}\mathbf{a}_p||\mathbf{a}_p||^2} \le \dfrac{\mathbf{a}_p^H\mathbf{R}^{-1}\mathbf{R}^{-1}\mathbf{a}_p||\mathbf{a}_p||^2}{(M-||\mathbf{a}_p||^2)\mathbf{a}_p^H\mathbf{R}^{-1}\mathbf{R}^{-1}\mathbf{a}_p+\mathbf{a}_p^H\mathbf{R}^{-1}\mathbf{R}^{-1}\mathbf{a}_p||\mathbf{a}_p||^2}=\dfrac{||\mathbf{a}_p||^2}{M}$$
As a consequence, one upper bound for $r$ is $\dfrac{||\mathbf{a}_p||^2}{M}$.
But through numerical tests, this bound is 10 times larger than the real values of $r$. The following figure is one of the numerical test results. The figure is drawn as a function of $\mathbf{a}$, or more specifically, as a function of $\theta$.
Can the ratio be tighter bounded?
Thank you.
P.S.,
1.Through simulations, it is found that $r$ decreases as $M$ increases. Can this be explained mathematically?
2.This problems comes from array signal processing.