# An Overview on the Applications of Matrix Theory in Wireless Communications and Signal Processing
## 3 Matrix Applications in Signal Estimation Theory
### SVD for Least-Squares Based Estimation  
Section 2.1 pointed out the important role played by SVD in modeling, implementation and characterization of MIMO channels. In this new section, we will illustrate the benefits offered by SVD in solving least-squares (LS) estimation problems [13].  
Consider a linear estimation problem defined as follows:  
\[\mathbf{\mathbf{x}}=\mathbf{A}\mathbf{\theta}+\mathbf{\
u},\]  
where \(\mathbf{\mathbf{x}}\in\mathbb{R}^{N}\) is the observed data, \(\mathbf{\
u}\in\mathbb{R}^{N}\) stands for the additive noise, \(\mathbf{\mathbf{A}}\in\mathbb{R}^{N\times P}\) is the given linear interaction matrix, and \(\mathbf{\theta}\in\mathbb{R}^{P}\) is the unknown vector to be estimated.  
The LS estimator is defined as  
\[\hat{\mathbf{\theta}}_{\text{LS}}=\arg\min_{\mathbf{\theta}}||\mathbf{A}\mathbf{\theta}- \mathbf{\mathbf{x}}||_{2}^{2}, \tag{14}\]  
where \(||\cdot||_{2}\) stands for the Euclidean norm. If \(\mathbf{\mathbf{A}}\) is full column rank, i.e., \(\text{rank}(\mathbf{\mathbf{A}})=P\) and \(N\geq P\), it admits the closed form solution  
\[\hat{\mathbf{\theta}}_{\text{LS}}=(\mathbf{\mathbf{A}}^{H}\mathbf{\mathbf{A}})^{-1}\mathbf{ \mathbf{A}}^{H}\mathbf{\mathbf{x}}. \tag{15}\]  
However, in the case of \(N<P\), matrix \(\mathbf{\mathbf{A}}^{H}\mathbf{\mathbf{A}}\) is rank deficient and this solution is invalid. In this case, an infinite number of minimizers exist, and the set of these solutions is denoted by \(\Theta\). A unique solution for the LS problem can be enforced by selecting the minimum \(l_{2}\) (Euclidean) norm solution from the set \(\Theta\). Thus, one should add this constraint to (14):  
\[\begin{split}\theta_{\text{LS}}&=\arg\min||\hat\mathbf{{\theta}}_{\text{LS}}||_{2}^{2}\\ \hat{\mathbf{\theta}}_{\text{LS}}&\in\Theta.\end{split} \tag{16}\]  
The new optimization problem defined by (14) and (16) can be solved by means of SVD. In particular, due to the SVD of \(\mathbf{\mathbf{A}}\), i.e., \(\mathbf{\mathbf{A}}=\mathbf{U}\mathbf{\Sigma}\mathbf{\mathbf{V}  
\[\mathbf{\Sigma}=\left[\begin{array}{cc}\mathbf{\Sigma}_{1}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}\end{array}\right].\]  
Moreover, \(\mathbf{b}\) and \(\mathbf{y}\) can also be decomposed as \(\mathbf{b}=[\mathbf{b}_{1}^{T}\ \mathbf{b}_{2}^{T}]^{T}\) and \(\mathbf{y}=[\mathbf{y}_{1}^{T}\ \mathbf{y}_{2}^{T}]^{T}\), respectively. Hence, (17) resumes to  
\[||\mathbf{A}\mathbf{\theta}-\mathbf{x}||_{2}^{2}=||\mathbf{\Sigma}_{1}\mathbf{y}_{1}-\mathbf{b }_{1}||_{2}^{2}+||\mathbf{b}_{2}||_{2}^{2}. \tag{18}\]  
Therefore, the minimum-norm LS-solution that minimizes \(||\mathbf{A}\mathbf{\theta}-\mathbf{x}||_{2}^{2}\) is given by \(\mathbf{y}_{1}=\mathbf{\Sigma}_{1}^{-1}\mathbf{b}_{1}\) and \(\mathbf{y}_{2}=\mathbf{0}\). Alternatively,  
\[\mathbf{y}=\left[\begin{array}{cc}\mathbf{\Sigma}_{1}^{-1}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}\end{array}\right]\left[\begin{array}{c}\mathbf{b}_{1}\\ \mathbf{b}_{2}\end{array}\right]=\mathbf{\Sigma}^{\dagger}\mathbf{b},\]  
where the block matrix is denoted as \(\mathbf{\Sigma}^{\dagger}\). Finally, the LS-estimate of \(\mathbf{\theta}\) can be obtained by  
\[\mathbf{\theta}_{\text{LS}}=\mathbf{V}\mathbf{\Sigma}^{\dagger}\mathbf{b}=\mathbf{V}\mathbf{\Sigma}^{ \dagger}\mathbf{U}^{H}\mathbf{x}.\]