Editing Arnoldi iteration (section)

==Krylov subspaces and the power iteration==

An intuitive method for finding the largest (in absolute value) eigenvalue of a given ''m'' × ''m'' matrix <math>A</math> is the [[power iteration]]:  starting with an arbitrary initial [[vector space|vector]] <var>b</var>, calculate {{nowrap|''Ab'', ''A''<sup>2</sup>''b'', ''A''<sup>3</sup>''b'', ...}} normalizing the result after every application of the matrix ''A''. 

This sequence converges to the [[eigenvector]] corresponding to the eigenvalue with the largest absolute value, <math>\lambda_{1}</math>. However, much potentially useful computation is wasted by using only the final result, <math>A^{n-1}b</math>. This suggests that instead, we form the so-called ''Krylov matrix'':
:<math>K_{n} = \begin{bmatrix}b & Ab & A^{2}b & \cdots & A^{n-1}b \end{bmatrix}.</math>

The columns of this matrix are not in general [[orthogonal]], but we can extract an orthogonal [[basis (linear algebra)|basis]], via a method such as [[Gram–Schmidt process|Gram–Schmidt orthogonalization]].  The resulting set of vectors is thus an orthogonal basis of the ''[[Krylov subspace]]'', <math>\mathcal{K}_{n}</math>.  We may expect the vectors of this basis to [[Linear span|span]] good approximations of the eigenvectors corresponding to the <math>n</math> largest eigenvalues, for the same reason that <math>A^{n-1}b</math> approximates the dominant eigenvector.