# Still don't understand why the $\mathbf w$ can let all beamforming be orthogonal to one another

I saw orthogonal beamforming (OBFM) in this paper : https://ieeexplore.ieee.org/document/7959644

Let me just explain what does the paper said about OBFM First ,in the picture above,the channel is

$$\mathbf H_m=\mathbf B_m^{LOS} \mathbf A_m^{LOS}+\sum\limits ^L_{l=1}\mathbf B_l^{NLOS}\mathbf A_l^{NLOS}= a (\mathbf \theta_m) \mathbf A_m^{LOS}+\sum\limits ^L_{l=1} a(\mathbf \theta_{m,l})\mathbf A_l^{NLOS}$$ And the received signals at the BS after performing beam formation is

$$\mathbf y_{out,m}=\sqrt{P} \mathbf w \mathbf H \mathbf x + \mathbf n_s=\sqrt{P} \mathbf w_m \mathbf H_m \mathbf x_m + \sum\limits_{j \neq m}^M \sqrt{P} \mathbf w_m \mathbf H_j \mathbf x_j+\mathbf n_{s,m}$$

Honestly,if $$\mathbf w = \mathbf H^{-1},$$ it would be perfect,however,in the real world ,we all know that it is impossible to know $$\mathbf H$$ all the time.And it is better to know $$\mathbf B^{LOS}$$,so we can assume $$\mathbf w = (\mathbf B^{LOS})^{-1},$$ so the $$\mathbf B^{LOS}$$ must be nonsingular matrix

Unfortunately,However, if the system employs a higher number of antennas compared to the number of users, the matrix $$\mathbf B^{LOS}$$ cannot be directly inversed,so we will use the Moore-Penrose pseudoinverse to let

$$\mathbf w = (\mathbf B^{LOS})^+=(\mathbf B^{LOS})^*( \mathbf B^{LOS}(\mathbf B^{LOS})^* )^{-1}$$

If we design $$\mathbf w$$,we can let all beamforming are orthogonal to one another, and so users do not encounter interference.

My question are

1. Why if the system employs a higher number of antennas compared to the number of users, the matrix $$\mathbf B^{LOS}$$ cannot be directly inversed ?

2. It seems that all $$w_m$$ will be the same,at first,all beamforming are not orthogonal to one another,why do they multiply the same thing and become orthogonal to one another?Where do i misunderstand?

3. What is the formula of $$a (\mathbf \theta_m)$$ ?