= Arnoldi iteration
{c}
{title2=$AQ_k=Q_{k+1}\widetilde H_k$}
Starting with $q_1=b/\|b\|_2$, the Arnoldi iteration constructs an <orthonormal basis> of successive <Krylov subspaces>. At step $j$, orthogonalize $Aq_j$ against $q_1,\ldots,q_j$, and normalize the remainder as $q_{j+1}$ if it is nonzero. The coefficients form an <upper Hessenberg matrix> $\widetilde H_k$ and satisfy $AQ_k=Q_{k+1}\widetilde H_k$. A zero remainder means the current <Krylov subspace> is invariant under $A$.
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