Summary
Ann×n complex matrixB is calledparacontracting if ‖B‖2≦1 and 0≠x∈[N(I-B)]⊥⇒‖Bx‖2<‖x‖2. We show that a productB=B k B k−1 ...B 1 ofk paracontracting matrices is semiconvergent and give upper bounds on the subdominant eigenvalue ofB in terms of the subdominant singular values of theB i 's and in terms of the angles between certain subspaces. Our results here extend earlier results due to Halperin and due to Smith, Solomon and Wagner. We also determine necessary and sufficient conditions forn numbers in the interval [0, 1] to form the spectrum of a product of two orthogonal projections and hence characterize the subdominant eigenvalue of such a product. In the final part of the paper we apply the upper bounds mentioned earlier to provide an estimate on the subdominant eigenvalue of the SOR iteration matrix ℒω associated with ann×n hermitian positive semidefinite matrixA none of whose diagonal entries vanish.
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The work of this author was supported in part by NSF Research Grant No. MCS-8400879
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Nelson, S., Neumann, M. Generalizations of the projection method with applications to SOR theory for hermitian positive semidefinite linear systems. Numer. Math. 51, 123–141 (1987). https://doi.org/10.1007/BF01396746
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DOI: https://doi.org/10.1007/BF01396746