# Convex \((\alpha,\beta)\)-Generalized Contraction and Its Applications in Matrix Equations
## 5 Solutions to Linear Matrix Equation  
In the following section, we are primarily interested in the positive definite solution of matrix equation:  
\[U-\Gamma_{1}^{*}U\Gamma_{1}-\cdots-\Gamma_{m}U\Gamma_{m}=Q \tag{20}\]  
where \(\Gamma_{1},\ldots,\Gamma_{m}\) are arbitrary matrices of order \(n\times n\) and \(Q\) is a matrix of order \(n\times n\). We can define a mapping \(G\) on the set \(\mathcal{H}(n)\), which comprises all Hermitian matrices of size \(n\times n\), as follows:  
\[G(U)=Q+\sum\limits_{j=1}^{m}\Gamma_{m}^{*}U\Gamma_{m} \tag{21}\]  
The fixed points of \(G\) correspond to the solutions of Equation (20). Let \(\Gamma\) belong to the set \(\mathcal{M}(n)\), which comprises all matrices of size \(n\times n\). In this context, the norm of \(\Gamma\), denoted as \(\|\Gamma\|_{1}\), can be expressed as the sum of its singular values \(s_{j}(\Gamma)\) for \(j=1,2,\ldots,n\), that is, \(\|\Gamma\|_{1}=\sum\limits_{j=1}^{n}s_{j}(\Gamma)\). Now, consider a given matrix \(Q_{+}\) from the set \(\mathcal{P}(n)\), which consists of all positive definite matrices of size \(n\times n\). We can define the following norm:  
\[\|\Gamma\|_{1,Q_{+}}=\|Q_{+}^{\frac{1}{2}}\Gamma Q_{+}^{\frac{1}{2}}\|_{1}.\]  
It can be observed that, for any \(Q_{+}\in\mathcal{P}(n)\), the set \(\mathcal{H}(n)\), equipped with the norm \(\|.\|_{1,Q_{+}}\), forms a Banach space. Additionally, \(\mathcal{H}(n)\) possesses a partial order. To indicate this partial order, we use the notation \(U\geq V\) (or \(U>V\)) when \(U-V\geq 0\) (or \(U-V>0\)), respectively. Furthermore, we define the spectral norm, denoted by \(\|.\|\), as follows: \(\|\Gamma\|=\sqrt{\lambda_{+}(\Gamma^{*}\Gamma)}\), where \(\lambda_{+}(\Gamma^{*}\Gamma)\) represents the largest eigenvalue of the matrix \(\Gamma^{*}\Gamma\). Also, let \(I_{n}\) represent the identity matrix of size \(n\times n\).  
Lemma 1 ([2]): _Consider matrices \(\Gamma\) and \(B\), both of size \(n\times n\), such that \(\Gamma,B\geq 0\). Then_  
\[\operatorname{tr}(\Gamma B)\leq\|\Gamma\|\operatorname{tr}(B).\]  
Theorem 5.1: _Let \(Q\in\mathcal{P}(n)\), additionally, there exists a matrix \(Q_{+}\in\mathcal{P}(n)\) such that_  
\[\left\|\sum\limits_{j=1}^{m}Q_{+}^{\frac{1}{2}}\Gamma_{j}Q_{+}\Gamma_{j}^{ \frac{1}{2}}Q_{+}^{\frac{1}{2}}\right\|<1\]  
_Then,_  
1. _The mapping_ \(G\) _has a unique fixed point in_ \(\mathcal{H}(n)\)_._
2. _Given an initial matrix_ \(U_{1}\in\mathcal{H}(n)\)_, the sequence_ \(U_{k}\) _is defined as follows:_ \[U_{k+1}=\frac{1}{2}U_{k}+\frac{1}{2}G(U_{k})\ \ \ \forall\ \ k\in\mathbb{N}\] _converges to the solution of (_20_), which lies within_ \(\mathcal{P}(n)\)Proof.: It can be observed that, for all \(U,V\in\mathcal{H}(n)\), there exists either a lower bound or an upper bound. When considering the case where \(U_{1}=0\), we find that \(G(0)=L(0)=Q>0\). Therefore, we have \(U_{1}\leq G(U_{1})\) and \(U_{1}\leq L(U_{1})\). Now, we shall demonstrate that the mapping \(G\) satisfies condition (10). For any matrix \(U,V\in\mathcal{H}(n)\) with \(U\leq V\), it follows that \(G(U)\leq G(V)\). Therefore,  
\[\left\|G\left(\frac{1}{2}U+\frac{1}{2}G(U)\right)-G(V)\right\|_{1,Q_{+}}=\left\|\sum_{j=1}^{m}\Gamma_{j}\bigg{(}\frac{1}{2}U+\frac{1}{2}G(U)-V \bigg{)}\Gamma_{j}^{*}\right\|_{1,Q_{+}}\] \[= \text{tr}\Bigg{\{}\sum_{j=1}^{m}Q_{+}^{\frac{1}{2}}\Gamma_{j}^{* }\Bigg{[}\frac{1}{2}(U-V)+\frac{1}{2}\Bigg{(}Q+\sum_{i=1}^{m}\Gamma_{i}^{*}U \Gamma_{i}-V\Bigg{)}\Bigg{]}\Gamma_{j}Q_{+}^{\frac{1}{2}}\Bigg{\}}\] \[= \sum_{j=1}^{m}\text{tr}\Bigg{\{}\Gamma_{j}Q_{+}\Gamma