## 1 Introduction  
In the paper, we revisit GMRES and GMRES(m) to solve  
\[\mathbf{A}\mathbf{x}=\mathbf{b},\;\mathbf{x},\mathbf{b}\in\mathbb{R}^{n},\; \mathbf{A}\in\mathbb{R}^{n\times n}. \tag{1}\]  
In terms of an initial guess, \(\mathbf{x}_{0}\), and the initial residual \(\mathbf{r}_{0}=\mathbf{b}-\mathbf{A}\mathbf{x}_{0}\), Equation (1) is equivalent to  
\[\mathbf{A}\mathbf{u}=\mathbf{r}_{0}, \tag{2}\]  
where  
\[\mathbf{u}=\mathbf{x}-\mathbf{x}_{0} \tag{3}\]  
is the descent vector or called a corrector. Equation (2) can be used to search a proper descent vector, \(\mathbf{u}\), which is inserted into Equation (3) to produce a solution, \(\mathbf{x}\), closer to the exact solution. We suppose an \(m\)-dimensional Krylov subspace:  
\[\mathcal{K}_{m}(\mathbf{A},\mathbf{r}_{0}):=\mathrm{Span}\{\mathbf{r}_{0}, \mathbf{A}\mathbf{r}_{0},\ldots,\mathbf{A}^{m-1}\mathbf{r}_{0}\}. \tag{4}\]Krylov subspaces are named after [PERSON] in a paper [1], who described a method for solving a secular equation to determine the frequencies of small vibrations of material systems. Many well-developed Krylov subspace methods for solving Equation (1) have been discussed in the text books [2; 3; 4; 5]. [PERSON] and [PERSON] [6] used an optimal representation of vectors in the Krylov space with the help of a variational problem; the extremum of the variational problem is the solution of the Kalman matrix equation, in which the 2-norm minimal solution is used as a characteristic of the Krylov space. One of the most important classes of numerical methods for iteratively solving Equation (1) is the Krylov subspace methods [7; 8], and the preconditioned Krylov subspace methods [9]. Recently, [PERSON] et al. [10] presented a comprehensive framework for unifying the Krylov subspace methods for Equation (1). The GMRES method in [11] used the Petrov-Galerkin method to search \(\mathbf{u}\in\mathcal{K}_{m}\) via a perpendicular property:  
\[\mathbf{r}_{0}-\mathbf{A}\mathbf{u}\perp\mathcal{L}_{m}=\mathbf{A}\mathcal{K }_{m}. \tag{5}\]  
The descent vector \(\mathbf{u}\in\mathcal{K}_{m}\) is achieved by minimizing the length of the residual vector [2]:  
\[\min\left\|\mathbf{r}\right\|, \tag{6}\]  
where  
\[\mathbf{r}=\mathbf{b}-\mathbf{A}\mathbf{x}=\mathbf{r}_{0}-\mathbf{A}\mathbf{ u}, \tag{7}\]  
and \(\left\|\mathbf{r}\right\|\) is the Euclidean norm of the residual vector, \(\mathbf{r}\). In many Krylov subspace methods, the central idea is the projection. The method of GMRES is a special case of the Krylov subspace methods with an oblique projection. Let \(\mathbf{V}\) and \(\mathbf{W}=\mathbf{A}\mathbf{V}\) be the matrix representations of \(\mathcal{K}_{m}\) and \(\mathcal{L}_{m}\), respectively. We project the residual Equation (2) from an \(n\)-dimensional space to an \(m\)-dimensional subspace:  
\[\mathbf{W}^{\mathrm{T}}\mathbf{A}\mathbf{u}=\mathbf{W}^{\mathrm{T}}\mathbf{r }_{0}. \tag{8}\]  
We seek \(\mathbf{u}\) in the subspace \(\mathcal{K}_{m}\) by  
\[\mathbf{u}=\mathbf{V}\mathbf{y}; \tag{9}\]  
hence, Equation (8) becomes  
\[\mathbf{W}^{\mathrm{T}}\mathbf{W}\mathbf{y}=\mathbf{W}^{\mathrm{T}}\mathbf{r }_{0}. \tag{10}\]  
Solving \(\mathbf{y}\) from Equation (10) and inserting it into Equation (9), we can obtain \(\mathbf{u}\), and, hence, the next \(\mathbf{x}=\mathbf{x}_{0}+\mathbf{u}\); \(\mathbf{x}\) is a correction of \(\mathbf{x}_{0}\), and we hope \(\left\|\mathbf{r}\right\|<\left\|\mathbf{r}_{0}\right\|\). Meanwhile, Equation (7) becomes  
\[\mathbf{A}\mathbf{u}=\mathbf{r}_{0}, \tag{2}\]0  
where \(\mathbf{W}\mathbf{y}=\mathbf{A}\mathbf{u}\). By means of Equations (5), (7), and (11), we enforce the perpendicular property by  
\[\mathbf{A}\mathbf{u}=\mathbf{r}_{0}, \tag{2}\]1  
it is equivalent to the projection of \(\mathbf{r}_{0}\) on \(\mathcal{L}_{m}\). \(\mathbf{W}\mathbf{y}=\mathbf{A}\mathbf{u}\) is the projection of \(\mathbf{r}_{0}\) on \(\mathcal{L}_{m}\), and \(\mathbf{r}\) is perpendicular to \(\mathcal{L}_{m}\), which ensures that the length of the residual vector is decreasing.