Starting with , the Arnoldi iteration constructs an orthonormal basis of successive Krylov subspaces. At step , orthogonalize against , and normalize the remainder as if it is nonzero. The coefficients form an upper Hessenberg matrix and satisfy . A zero remainder means the current Krylov subspace is invariant under .
For a real symmetric matrix , the leading block of the Arnoldi decomposition is both symmetric and an upper Hessenberg matrix, hence a tridiagonal matrix. The resulting three-term recurrence is the Lanczos algorithm. If is a symmetric positive-definite matrix, its compression is too, since for nonzero .
The factorization produced by the Arnoldi iteration, with orthonormal and an upper Hessenberg matrix . Multiplying by identifies its leading square block as .
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Arnoldi iteration is an important numerical method used in linear algebra for approximating the eigenvalues and eigenvectors of a large, sparse matrix. It is particularly useful for solving problems in fields such as scientific computing, quantum mechanics, and engineering, where one may encounter large systems that cannot be solved directly due to computational limitations. ### Overview The Arnoldi iteration algorithm builds an orthonormal basis for the Krylov subspace generated by the matrix in question.