# An Overview on the Applications of Matrix Theory in Wireless Communications and Signal Processing
## 3 Matrix Applications in Signal Estimation Theory
### Matrix Inverse Lemma for Derivation of the Recursive Least-Squares Filter
As we described earlier in (14) and (15), the problem (21) admits the solution  
\[\hat{\mathbf{h}}_{n}=(\mathbf{A}_{n}^{H}\mathbf{A}_{n})^{-1}\mathbf{A}_{n}^{ H}\mathbf{x}_{n}.\]  
Setting \(\mathbf{R}_{n}=\mathbf{A}_{n}^{H}\mathbf{A}_{n}\) and \(\mathbf{p}_{n}=\mathbf{A}_{n}^{H}\mathbf{x}_{n}\) leads to  
\[\hat{\mathbf{h}}_{n}=\mathbf{R}_{n}^{-1}\mathbf{p}_{n}. \tag{22}\]  
Moreover, since  
\[\mathbf{R}_{n}=\mathbf{A}_{n}^{H}\mathbf{A}_{n}=\sum_{i=1}^{n}\mathbf{a}_{i} \mathbf{a}_{i}^{H}=\sum_{i=1}^{n-1}\mathbf{a}_{i}\mathbf{a}_{i}^{H}+\mathbf{a }_{n}\mathbf{a}_{n}^{H}=\mathbf{R}_{n-1}+\mathbf{a}_{n}\mathbf{a}_{n}^{H},\]  
(22) can be expressed as  
\[\hat{\mathbf{h}}_{n}=(\mathbf{R}_{n-1}+\mathbf{a}_{n}\mathbf{a}_{n}^{H})^{-1 }\mathbf{p}_{n}. \tag{23}\]  
Applying the matrix inverse lemma (19) to the inverse of the matrix contained in Equation (23) yields:  
\[\mathbf{R}_{n}^{-1}=(\mathbf{R}_{n-1}^{-1}+\mathbf{a}_{n}\mathbf{a}_{n}^{H})^ {-1}=\mathbf{R}_{n-1}^{-1}-\frac{\mathbf{R}_{n-1}^{-1}\mathbf{a}_{n}\mathbf{ a}_{n}^{H}\mathbf{R}_{n-1}^{-1}}{1+\mathbf{a}_{n}^{H}\mathbf{R}_{n-1}^{-1} \mathbf{a}_{n}}. \tag{24}\]  
Furthermore, defining \(\mathbf{k}_{n}=\frac{\mathbf{R}_{n-1}^{-1}\mathbf{a}_{n}}{1+\mathbf{a}_{n}^{H }\mathbf{R}_{n-1}^{-1}\mathbf{a}_{n}}\) and \(\mathbf{\Sigma}_{n}=\mathbf{R}_{n}^{-1}\), it follows that  
\[\mathbf{\Sigma}_{n}=\mathbf{\Sigma}_{n-1}-\mathbf{k}_{n}\mathbf{a}_{n}^{H} \mathbf{\Sigma}_{n-1}, \tag{25}\]  
and  
Figure 4: RLS filter. \[\Sigma_{n}\mathbf{a}_{n} =\Sigma_{n-1}\mathbf{a}_{n}-\mathbf{k}_{n}\mathbf{a}_{n}^{H}\Sigma_{ n-1}\mathbf{a}_{n}\] \[=\mathbf{k}_{n}(1+\mathbf{a}_{n}^{H}\Sigma_{n-1}\mathbf{a}_{n})- \mathbf{k}_{n}\mathbf{a}_{n}^{H}\Sigma_{n-1}\mathbf{a}_{n} \tag{26}\] \[=\mathbf{k}_{n}.\]  
Plugging (25) and (26) into (22) results in  
\[\hat{\mathbf{h}}_{n}=\mathbf{R}_{n}^{-1}\mathbf{p}_{n}. \tag{22}\]0 \[\hat{\mathbf{h}}_{n}=\mathbf{R}_{n}^{-1}\mathbf{p}_{n}. \tag{22}\]1 \[\hat{\mathbf{h}}_{n}=\mathbf{R}_{n}^{-1}\mathbf{p}_{n}. \tag{22}\]2 \[\hat{\mathbf{h}}_{n}=\mathbf{R}_{n}^{-1}\mathbf{p}_{n}. \tag{22}\]3 \[\hat{\mathbf{h}}_{n}=\mathbf{R}_{n}^{-1}\mathbf{p}_{n}. \tag{22}\]4 \[\hat{\mathbf{h}}_{n}=\mathbf{R}_{n}^{-1}\mathbf{p}_{n}. \tag{22}\]5 \[\hat{\mathbf{h}}_{n}=\mathbf{R}_{n}^{-1}\mathbf{p}_{n}. \tag{22}\]6  
where \(\epsilon(n)=x(n)-\mathbf{a}_{n}^{H}\hat{\mathbf{h}}_{n-1}\) and the second equality comes from the fact that  
\[\hat{\mathbf{h}}_{n}=\mathbf{R}_{n}^{-1}\mathbf{p}_{n}. \tag{22}\]7  
In a nutshell, the RLS algorithm can be summarized as follows:  
1. Choose a large positive number \(\alpha\), and initialize \(\mathbf{\Sigma}_{0}=\alpha\mathbf{I}\) and \(\hat{\mathbf{h}}_{0}=\mathbf{0}\). 2. For \(n=1,2,\cdots,N\), sequentially update \[\hat{\mathbf{h}}_{n}=\mathbf{R}_{n}^{-1}\mathbf{p}_{n}. \tag{22}\]8 \[\hat{\mathbf{h}}_{n}=\mathbf{R}_{n}^{-1}\mathbf{p}_{n}. \tag{22}\]9 \[\mathbf{R}_{n}=\mathbf{A}_{n}^{H}\mathbf{A}_{n}=\sum_{i=1}^{n}\mathbf{a}_{i} \mathbf{a}_{i}^{H}=\sum_{i=1}^{n-1}\mathbf{a}_{i}\mathbf{a}_{i}^{H}+\mathbf{a }_{n}\mathbf{a}_{n}^{H}=\mathbf{R}_{n-1}+\mathbf{a}_{n}\mathbf{a}_{n}^{H},\]0 \[\mathbf{R}_{n}=\mathbf{A}_{n}^{H}\mathbf{A}_{n}=\sum_{i=1}^{n}\mathbf{a}_{i} \mathbf{a}_{i}^{H}=\sum_{i=1}^{n-1}\mathbf{a}_{i}\mathbf{a}_{i}^{H}+\mathbf{a }_{n}\mathbf{a}_{n}^{H}=\mathbf{R}_{n-1}+\mathbf{a}_{n}\mathbf{a}_{n}^{H},\]1  
As the algorithm processes, the error made by approximating \(\mathbf{\Sigma}_{0}=\alpha\mathbf{I}\) and \(\hat{\mathbf{h}}_{0}=\mathbf{0}\) becomes less and less significant [12]. In some cases, a weighted least squares estimation error is used as the minimizer by multiplying the error terms with so-called \"forgetting factors\", which are intended to reduce the influence of old data [14]. The RLS algorithm has emerged as a powerful tool for adaptive filtering, prediction and system identification applications.