Symbols

      $A, \ldots, Z$ matrices 

$a, \ldots, z$ vectors       

$\alpha, \beta, \ldots, \omega$ scalars 

$A^{T}$ matrix transpose 

$A^{\ast}$ conjugate transpose of $A$ 

$A^{-1}$ matrix inverse 

$A^{-T}$ the inverse of $A^T$ 

$A^{-\ast}$ the inverse of $A^*$ 

$A^{m \times n}$ indicates that $A$ is $m$-by-$n$ 

$a_{ij}$ matrix element of $A$ 

$a_{i,.}$ $i$th row of matrix $A$

$a_{.,j}$ $j$th column of matrix $A$ 

$x^{\ast}y$ vector dot product  

$(x,y)$ inner product, such as $x^{\ast}y$ 

$v_j$ vector $v$ in the $j$th iteration


diag($A$) 		 diagonal of matrix $A$ 


diag($\alpha, \beta, \ldots$) 		 diagonal matrix constructed                          from scalars $\alpha, \beta, \ldots$


span($a,b, \ldots$) 		 spanning space of vectors $a,b, \ldots$ 


span($A$) 		 $= \mbox{range}(A)$,            subspace spanned by the columns  of $A$ 

${\cal K}^{m}(A,v)$ Krylov subspace         $\mbox{span}\{ v,Av, A^2 v, \ldots, A^{m-1} v \}$ 

$[x_1,x_2, \ldots ,x_m]$ matrix whose $i$th column is $x_i$ 

$X(i, j)$ $(i, j)$ entry of $X$

$X(:,i)$ or $X(i,:)$ $i$th column or row of $X$ 

$X(:,[2,3,5])$ submatrix consisting of columns 2, 3, and 5 of $X$ 

$X([2,3,5],:)$ submatrix consisting of rows 2, 3, and 5 of $X$ 

$\cal R$, $\cal C$ sets of real and complex numbers 

${\cal R}^{n}$, ${\cal C}^n$ real and complex $n$-spaces

$\Vert x \Vert _{p}$, $\Vert A \Vert _p$ vector and matrix $p$-norm 

$\Vert x \Vert _{A}$ the ``$A$-norm,'' defined as $(Ax,x)^{1/2}$ 

$\Vert A \Vert _F$ matrix Frobenius norm 

$\epsilon_M$ machine precision 

$\lambda(A)$ eigenvalues of $A$ 

$\lambda_{\max}(A), \lambda_{\min}(A)$ eigenvalues of $A$ with                          maximum (resp., minimum) modulus 

$\sigma(A)$ singular values of $A$ 

$\sigma_{\max}(A), \sigma_{\min}(A)$ largest and smallest                          singular values of $A$ 

$\kappa_2 (A)$ condition number of matrix $A$, defined as                          $\kappa_2 (A) = \Vert A\Vert _2\Vert A^{-1}\Vert _2$ 

$\overline{\alpha}$ complex conjugate of the scalar $\alpha$ 

$\max \{S\}$ maximum value in set $S$ 

$\min \{S\}$ minimum value in set $S$ 

$\sum$ summation 

$O(\cdot)$ ``big-oh'' asymptotic bound 

$\re(a)$ the real part of the complex number $a$ 

$\im(a)$ the imaginary part of the complex number $a$ 

$u \perp v$ the vector $u$ is orthogonal to the vector $v$