As convergence takes place, the subdiagonals of tend to zero and the most desired eigenvalue approximations appear as eigenvalues of the leading block of in a Schur decomposition of . The basis vectors tend to orthogonal Schur vectors.
An alternate interpretation stems from the fact that each of these shift cycles results in the implicit application of a polynomial in of degree to the starting vector. v_1 (A) v_1 with () = _j=1^p (- _j ). The roots of this polynomial are the shifts used in the QR process and these may be selected to enhance components of the starting vector in the direction of eigenvectors corresponding to desired eigenvalues and damp the components in unwanted directions. Of course, this is desirable because it forces the starting vector into an invariant subspace associated with the desired eigenvalues. This in turn forces to become small and hence convergence results. Full details may be found in .
There is a fairly straightforward intuitive explanation
of how this repeated updating of the starting
through implicit restarting might lead to convergence.
If is expressed as a linear combination of eigenvectors
of , then
It is worth noting that if then and this iteration is precisely the same as the implicitly shifted QR iteration. Even for , the first columns of and the Hessenberg submatrix are mathematically equivalent to the matrices that would appear in the full implicitly shifted QR iteration using the same shifts In this sense, the implicitly restarted Arnoldi method may be viewed as a truncation of the implicitly shifted QR iteration. The fundamental difference is that the standard implicitly shifted QR iteration selects shifts to drive subdiagonal elements of to zero from the bottom up while the shift selection in the implicitly restarted Arnoldi method is made to drive subdiagonal elements of to zero from the top down. Shifted power-method-like convergence will be obtained.
When the exact shift strategy is used, implicit restarting can be viewed both as a means to damp unwanted components from the starting vector and also as directly forcing the starting vector to be a linear combination of wanted eigenvectors. See  for information on the convergence of IRAM and [22,421] for other possible shift strategies for Hermitian The reader is referred to [292,333] for studies comparing implicit restarting with other schemes.