Theory vs experiment in matrix analysis

Guided by theory, we use visualization and numerical experiment to investigate two problem areas in matrix analysis. One is the question of tridiagonability of matrices by unitary similarity. We raise a more general question concerning the simultaneous Hessenberg form of a pair of matrices. The critical dimension is 4 for both tridiagonability and simultaneous Hessenberg form. Our findings in the tridiagonable case are borne out by a recent theorem of V. Pati. We obtain the simultaneous Hessenberg forms in certain special cases. An algorithm that finds the two forms numerically is used to strengthen the conjecture that every pair of matrices has simultaneous Hessenberg form. The algorithm is also used to exhibit a simple 5 x 5 matrix that is not tridiagonable. The second question we address concerns the two inequalities 'w'(' AB') <= 'w'('A')'w'(' B') and 'w'('A')'w'(' B') <= 'w'('A')

'B'

for commuting operators 'A' and 'B', where w is the numerical radius. We show that a Pick condition on the two operators is sufficient to ensure that the inequalities hold. The theory of 0-1 matrices combined with numerical work has revealed a new class of counterexamples to the secand inequality for 7 x 7 matrices. Until this work such counterexamples were known only in dimensions 9 and above. Moreover, a genetic algorithm has even found weak counterexamples using two general commuting 4 x 4 matrices.