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The extended Davis-Kahan theorem makes use of polynomial matrixtransformations to produce bounds at least as tight as the standard Davis-Kahantheorem. The optimization problem of finding transformation parametersresulting in optimal bounds from the extended Davis-Kahan theorem is presentedfor affine transformations. It is demonstrated how globally optimal boundvalues can be computed automatically using fractional programming theory. Twodifferent solution approaches, the Charnes-Cooper transformation andDinkelbach's algorithm are reviewed. Our implementation of the extendedDavis--Kahan theorem is used to calculate bound values in three significantexamples. First, a pairwise comparison is made of the spaces spanned by theeigenvectors of the graph shift operator matrices corresponding to differentstochastic block model graphs. Second our bound is calculated on the distanceof the spaces spanned by eigenvectors of the graph shift operators and theircorresponding generating matrices in the stochastic blockmodel, and, third, onthe sample and population covariance matrices in a spiked covariance model. Ourextended bound values, using affine transformations, not only outperform thestandard Davis-Kahan bounds in all examples where both theorems apply, but alsodemonstrate good performance in several cases where the standard Davis-Kahantheorem cannot be used.

Extending the Davis-Kahan theorem for comparing eigenvectors of two symmetric matrices II: Computation and Applications