The function is given as follows: $\hat{\mathbf{b}}^{H}\left ( \boldsymbol{\theta} \right ) = \left ( \mathbf{b_{u}} + \mathbf{A_{u}}diag \left ( \mathbf{d_{u}} \right )\boldsymbol{\theta} \right ) \left ( \mathbf{b^{H}_{d}} + \mathbf{\boldsymbol{\theta}}^{T}diag \left ( \mathbf{d_{d}} \right ) \mathbf{A^{H}_{d}} \right ) = \hat{\mathbf{b}_{u}}\left ( \boldsymbol{\theta} \right ) \hat{\mathbf{b}_{d}}^{H}\left ( \boldsymbol{\theta} \right ) $ , where all are fixed except $\boldsymbol{\theta}$
Now the main function is $ P= \left \| \hat{\mathbf{b}}^{H}\left ( \boldsymbol{\theta} \right )\mathbf{v} \right \|^{2}$ is convex , non-convex, concave and non-concave ?? Here, I am trying to calculate the hessian for 2 variable (Not general case). But in general, how to determine the convexity of the given function? Please help me this regard? Why not general solution to verify convexity of this type of functions rather than trail and error methods?
Where,
$ \mathbf{A_{u}} \in \mathbb{C}^{M_{r}\times N}$ ,
$ \mathbf{b_{d}} \in \mathbb{C}^{M_{t}\times 1}$,
$ \mathbf{A_{d}} \in \mathbb{C}^{M_{t}\times N}$,
$ \mathbf{d_{d}} \in \mathbb{C}^{N\times 1}$,
$ \mathbf{b_{u}} \in \mathbb{C}^{M_{r}\times 1}$,
$ \mathbf{d_{u}} \in \mathbb{C}^{N\times 1}$ ,
$\mathbf{v} \in \mathbb{C}^{M_{t}\times 1}$,
$\boldsymbol{\theta} \in \mathbb{C}^{N\times 1}$