## 3 Problem Formulation and Solution
The gradient of \(R_{ij}(\boldsymbol{\theta},\mathbf{w})\)is defined as \(\
abla R_{ij}(\boldsymbol{\theta},\mathbf{w})=[\
abla_{\theta}R_{ij}( \boldsymbol{\theta},\mathbf{w});\
abla_{\eta}R_{ij}(\boldsymbol{\theta}, \mathbf{w})]\). As shown in Equation (8), the transmission rate \(R_{ij}(\boldsymbol{\theta},\mathbf{w})\) has a complicated form, which is mainly caused by the SINR part. There are a series of summations of variables,making the derivative of \(R_{ij}(\mathbf{\theta},\mathbf{w})\) hard to express. To deal with this, we bring out a method to convert the summations into matrix multiplications and subsequently derive the gradient for applying the linear approximation. We reconstruct the optimization variables by introducing a series of vectors and matrices. Let  
\[\mathbf{w}=\left[\underbrace{\underbrace{\mathbf{w}_{11},\mathbf{w}_{12}, \ldots\ldots,\mathbf{w}_{1j},\ldots\ldots,\mathbf{w}_{1N},\underbrace{\mathbf{ w}_{21},\ldots\ldots,\mathbf{w}_{2N},}_{N\cdot N_{T}},\ldots\ldots,\mathbf{w}_{ ij},\ldots\ldots,\mathbf{w}_{MN}}_{M\cdot N\cdot N_{T}}}_{M\cdot N\cdot N_{T}} \right]^{T}. \tag{16}\]  
As one of optimization variables in this problem, \(\mathbf{w}\) is a vector variable with \(M\cdot N\cdot N_{T}\) dimensions, contains all beamforming weight information. The expression of SINR in matrix form can be constructed as follows:  
\[\mathbf{\hat{H}}_{ij}=\mathbf{h}_{ij}\mathbf{h}_{ij}^{H},i\in\mathcal{M},j \in\mathcal{N},\text{ and then, let}\]  
\[\mathbf{\hat{H}}_{ij}=Bdiag\left(\underbrace{\underbrace{\mathbf{\hat{H}}_{ ij}\text{th block is }\mathbf{0}}_{N\text{ blocks}},\underbrace{\mathbf{\hat{H}}_{ij},\ldots,\mathbf{0}, \ldots,\mathbf{\hat{H}}_{1j},}_{N\text{ blocks}},\underbrace{\mathbf{\hat{H}}_{ 2j},\ldots,\mathbf{0},\ldots,\mathbf{\hat{H}}_{2j}}_{N\text{ blocks}},\ldots, \underbrace{\mathbf{\hat{H}}_{0},\ldots,\mathbf{0}}_{N\text{ blocks}},\ldots, \underbrace{\mathbf{\hat{H}}_{Mj},\ldots,\mathbf{0},\ldots,\mathbf{\hat{H}}_{ MJ}}_{N\text{ blocks}}}_{M\cdot N\text{ blocks}},\ldots,\underbrace{\mathbf{\hat{H}}_{Mj},\ldots,\mathbf{\hat{H}}_{ MJ}}_{N\text{ blocks}}}_{N\text{ blocks}}\right). \tag{17}\]  
Note that every \(\mathbf{\hat{H}}_{ij}\) is a \(N_{T}\)-dimension square matrix and that \(\mathbf{\hat{H}}_{ij}\) is a block diagonal matrix with \(M\cdot N\) blocks. Specifically, the diagonal blocks of \(\mathbf{\hat{H}}_{ij}\) are divided into \(M\) groups, and each group contains \(N\) blocks. Every diagonal block is a \(N_{T}\times N_{T}\) matrix, so we can see that each \(\mathbf{\hat{H}}_{ij}\) is a square matrix with \(M\cdot N\cdot N_{T}\) dimensions. Actually, \(\mathbf{\hat{H}}_{ij}\) consists of a series of \(\mathbf{\hat{H}}_{ij}\) and zero matrices. The \(N\) blocks of a group are all nearly the same, except the \(j\)th block of every group is a zero matrix. Furthermore, the \(N\) blocks of the \(i\)th group are all zero matrices. Except those above blocks, the blocks in the \(i^{*}\)th group are set as \(\mathbf{\hat{H}}_{i^{*}j}\). The blocks of the \(i^{*}\)th group roughly represent the channels between interference source UAV\({}_{i^{*}}\) and UE\({}_{j}\). For the downlink channel from UAV\({}_{i}\) to UE\({}_{j}\), the interference caused by signals sent from other UAVs to other users can be represented by \(\mathbf{w}_{ij}^{H}\mathbf{\hat{H}}_{ij}(\mathbf{\theta})\mathbf{w}_{ij}\). Let  
\[\mathbf{H}_{ij}= \tag{18}\]  
\[\mathbf{H}_{ij}\] is almost the same as \[\mathbf{\hat{H}}_{ij}\] ; the only difference is the \[j\]th block in the \[i\]th group.