Obtaining Bounds on the Two Norm of a Matrix from the Splitting Lemma

Dedicated to Alan George on the occasion of his 60th birthday Abstract. The splitting lemma is one of the two main tools of support theory, a framework for bounding the condition number of definite and semidefinite preconditioned linear systems. The splitting lemma allows the analysis of a complicated system to be partitioned into analyses of simpler systems. The other tool is the symmetric-productsupport lemma, which provides an explicit spectral bound on a preconditioned matrix. The symmetric-product-support lemma shows that under suitable conditions on the null spaces of and , the finite eigenvalues of the pencil are bounded by , where , , and . To apply the lemma, one has to construct a satisfying these conditions, and to bound its -norm. In this paper we show that in all its existing applications, the splitting lemma can be viewed as a mechanism to bound for a given . We also show that this bound is sometimes tighter than other easily-computed bounds on , such as and