Structure Preserving Dynamical Systems for Matrices – A Revisit and Some New Discoveries

The notion of structure preserving has been put into practice in numerical linear algebra since its very early stage of development. For example, the upper Hessenberg form in the QR algorithm, the upper Hessenberg/triangular form in the the QZ algorithm, and the bidiagonal form in the SVD algorithm, not only are preserved throughout the iter-ative process, but also play a fundamental role in making these algorithms effective for computation. Each of these iterative schemes has a corresponding continuous analogue which was designed initially to preserve the above-mentioned structure. Only recently it is discovered that these continuous dynamical systems and hence perhaps together with their original discrete iterative schemes preserve some additional structures — notably the staircase structure. This fact, though not necessarily of practical value, is mathematically intriguing due to the reticence it has held and the complexity its proof would have in-volved. This talk surveys some of the existing flows, outlines the mathematics involved in the structures preserving, and generalizes some of the ideas to matrix pencils wit