QR algorithm is one of the major methods in computing the complete eigenpairs of a dense matrix.It is well known that QR algorithm is a smart realization of simultaneous iteration which is a practical method in carrying on subspace iteration.So

the convergence of subspace iteration dwells in the central part of this area.Previous proofs always assume that matrix A has a complete orthogonal eigensystem

while in this paper

the convergence

similar to that of power method

is established without requiring A to meet any special condition.On this basis

we explain how QR iteration converges to a block upper triangular matrix

and then state briefly the relationship between QR algorithm and simultaneous iteration.=