Given a vector $w\in \mathcal{C}^{M\times 1}$ and a constant $a$, we can have many vectors $u \in \mathcal{C}^{M\times 1}$, such that $u^H u= 1$ and $|u^H w| = a$. Suppose we have uniform distribution of the unitary $u$, what is the expectation $E(u\times u^H)$ of such $u$ that satisfies the above equations?
I have tried the following method. Let $$u= a \times e^{j\theta} \times w + \sum_{i=1}^{M-1} b_j w_j^0$$ where $w_j^0$ is the base vectors if the null space of $w$, and $b_j$ is random variables such that $\sum_{j=1}^{M-1} |b_j|^2=1-a^2$. Therefore, \begin{eqnarray} E(uu^H) &= & a^2 w w^H + \sum_{i=1}^{M-1}E( |b_i|^2 )w_i^0 w_i^{0H} \\ &= & a^2 w w^2 + \frac{1-a^2}{M-1}\sum_{i=1}^{M-1}w_i^0 w_i^{0H} \\ &= & a^2 w w^H + \frac{1-a^2}{M-1} (I - w w^H)\\ &=& \frac{1-a^2}{M-1} I + \frac{M a^2 - 1}{M-1} w w^H \end{eqnarray}
Please have a look whether it is correct or not. Thanks a lot.