## 3 A Double Optimal Solution
### Mathematical Preliminaries
In the proof of Theorem 1, we have used the minimality condition of Equation (40) to determine \(\mathbf{\alpha}\) in Equation (62), and then the minimality condition of Equation (41) was used to determine \(\alpha_{0}\) in Equation (71). Two optimizations were used to derive \(\mathbf{x}\) in Equation (42), which is thus named a double optimal solution (DOS). Theorem 2: _The two factors \(\lambda_{0}\) in Equation (43) and \(\alpha_{0}\) in Equation (45) satisfy_  
\[\mathbf{w}=\lambda_{0}\mathbf{P}\mathbf{b}+\mathbf{A}\mathbf{u}_ {0}-\mathbf{P}\mathbf{A}\mathbf{u}_{0},\]2  
Proof. : It follows from Equations (59) and (56) that  
\[\mathbf{w}=\lambda_{0}\mathbf{P}\mathbf{b}+\mathbf{A}\mathbf{u}_ {0}-\mathbf{P}\mathbf{A}\mathbf{u}_{0},\]3  
hence, \(\mathbf{P}\) is a projection operator. In Equation (44), \(\mathbf{w}\) can be written as  
\[\mathbf{w}=\lambda_{0}\mathbf{P}\mathbf{b}+\mathbf{A}\mathbf{u}_ {0}-\mathbf{P}\mathbf{A}\mathbf{u}_{0},\]4  
Utilizing Equation (73), \(\mathbf{P}^{\mathrm{T}}=\mathbf{P}\), and \((\mathbf{I}_{q}-\mathbf{P})^{2}=\mathbf{I}_{q}-\mathbf{P}\), it follows that  
\[\mathbf{w}=\lambda_{0}\mathbf{P}\mathbf{b}+\mathbf{A}\mathbf{u}_ {0}-\mathbf{P}\mathbf{A}\mathbf{u}_{0},\]5 \[\mathbf{w}=\lambda_{0}\mathbf{P}\mathbf{b}+\mathbf{A}\mathbf{u}_ {0}-\mathbf{P}\mathbf{A}\mathbf{u}_{0},\]6  
With the aid of Equation (65), Equation (76) is further reduced to  
\[\mathbf{w}=\lambda_{0}\mathbf{P}\mathbf{b}+\mathbf{A}\mathbf{u}_ {0}-\mathbf{P}\mathbf{A}\mathbf{u}_{0},\]7  
Now, after inserting Equation (77) into Equation (71), we can obtain  
\[\mathbf{w}=\lambda_{0}\mathbf{P}\mathbf{b}+\mathbf{A}\mathbf{u}_ {0}-\mathbf{P}\mathbf{A}\mathbf{u}_{0},\]8  
which is just equal to \(\|\mathbf{w}\|^{2}\) by Equation (75). Hence, we obtain Equation (72) readily. The proof of \(\alpha_{0}\lambda_{0}=1\) is complete. The DOS is derived in the VAKS \(\mathcal{K}_{m}^{\prime}\) in a way to stick onto Equation (21). If one takes \(\alpha_{0}=1\) in Equation (21), \(\mathcal{K}_{m}^{\prime}\) reduces to the usual affine Krylov subspace \(\mathbf{u}_{0}+\mathcal{K}_{m}\). However, in this situation, Theorem 1 no longer holds. If one takes \(\mathbf{u}_{0}=\mathbf{0}\) and works directly in the \((m+1)\)-dimensional Krylov \(\mathcal{K}_{m+1}\) by expanding \(\boldsymbol{\alpha}\) in an \((m+1)\)-dimensional subspace, it would result in failure, because after inserting \(\mathbf{y}_{0}=\mathbf{Au}_{0}=\mathbf{0}\) into Equation (62), \(\boldsymbol{\alpha}\) cannot be defined. We can conclude that the DOS directly benefits from the VAKS. The following corollaries are proven to help us understand the properties of DOS whose bases appeared in Theorems 1 and 2 and equations appeared in the proof of Theorem 1. The multiplier \(\lambda\) defined by Equation (61) satisfies  
\[\mathbf{w}=\lambda_{0}\mathbf{P}\mathbf{b}+\mathbf{A}\mathbf{u}_ {0}-\mathbf{P}\mathbf{A}\mathbf{u}_{0},\]9  
and the double optimal solution (42) can be written as  
\[\alpha_{0}=\frac{\mathbf{w}^{\mathsf{T}}\mathbf{b}}{\|\mathbf{w} \|^{2}}. \tag{45}\]0  
Inserting Equation (39) for \(\mathbf{y}_{0}\) into Equation (61), we have  
\[\alpha_{0}=\frac{\mathbf{w}^{\mathsf{T}}\mathbf{b}}{\|\mathbf{w} \|^{2}}. \tag{45}\]1  
in view of Equation (65), which can be written as  
\[\alpha_{0}=\frac{\mathbf{w}^{\mathsf{T}}\mathbf{b}}{\|\mathbf{w} \|^{2}}. \tag{45}\]2  
Hence, \(\lambda=1\) follows immediately from Equation (72). Then, using Equations (72) and (42) proves Equation (80). Inserting Equation (20) for \(\mathbf{u}_{0}\) into Equation (80) yields  
\[\alpha_{0}=\frac{\mathbf{w}^{\mathsf{T}}\mathbf{b}}{\|\mathbf{w} \|^{2}}. \tag{45}\]3  
which is a pseudo-inverse solution obtained from the DOS. **Corollary 2**.