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{\topskip=100pt \noindent \LARGE \hfil List of Symbols \hfil\par}

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    \begin{tabular}{ll}

    $A,\cdots,Z$                                 & matrices \\   

    $\textit{\bfseries a},\cdots,\textit{\bfseries z}$                                 & vectors \\

    $\alpha,\beta,\cdots,\omega$                 & scalars \\
    
    $A^{\mathrm H}$                                      & conjugate transpose (Hermitian) of A \\
    
    $A^{-1}$                                     & matrix inverse \\

    $a_{ij}$                                     & matrix element \\

    $a_{i}$                                      & vector element \\
    
    ${\cal C}$                                      & set of complex numbers \\

    ${\cal C}^{n}$                             & complex $n$-space \\
    
    $\textit{\bfseries x}_{n}$                                      & vector $\textit{\bfseries x}$ in the $n$th iteration \\
    
    $(\textit{\bfseries x},\textit{\bfseries y})$                                      & vector dot product (inner product), $\textit{\bfseries x}^{\mathrm{H}}\textit{\bfseries y}=\sum_{i=1}^{N} \bar{x}_{i}y_{i}$ \\
    
    diag$(\alpha,\beta,\cdots)$                  & diagonal matrix constructed from scalars $\alpha,\beta,\cdots$ \\
    
    Span$\{\textit{\bfseries a},\textit{\bfseries b},\cdots\}$                         & spanning space of vector $\textit{\bfseries a},\textit{\bfseries b},\cdots$ \\
    
    $K_{n}(A;\textit{\bfseries r}_{0})$                             & Krylov subspace, defined as span$\{{\textit{\bfseries r}_{0},A\textit{\bfseries r}_{0},\cdots,A^{n-1}\textit{\bfseries r}_{0}}\}$ \\
    
    $\cal R$                                          & set of real numbers \\

	${\cal R}^{n}$                                      & real $n$-space \\
	
	$\parallel \textit{\bfseries x} \parallel$                      & 2-norm \\
	
	$\parallel \textit{\bfseries x} \parallel_{A}$                & the "A-norm", defiend as $(A\textit{\bfseries x},\textit{\bfseries x})^{1/2}$ \\
	
	$\kappa(A)$                                 & spectral condition number of matrix A \\
	
	$\rho(A)$                                     & spectrum of matrix A \\
	
	$\bar{\alpha}$                               & complex conjugate of the matrix \\
	
	$D$                                            & diagonal matrix \\
	
	$L$                                             & (Strict) lower triangular matrix \\
	
	$U$                                             & (Strict) upper triangular matrix \\
	
	$H$                                             & Hessenberg matrix \\
	
	$R_{n}(\lambda)$                            & Lanczos polynomial of the degree $n$ \\

	$\tilde{A}$                                   & new coefficient matrix of the linear system \\
	
	$\lambda_{ij}$                                & eigenvalues \\
	
	$\textit{\bfseries u}_{ij}$                                         & eigenvectors corresponding to $\lambda_{ij}$
    \end{tabular}