Modern Matrix Computations and Applied Mathematics, a Survey

The sole purpose of this Math Survey paper is to describe my recent work that helped to solve a cen- tury old Quantum Physics problem of Einstein, Neu- mann, Bohr, and Wigner. This fundamental Quan- tum result had lain open and unsolved since the 1920s. My solution depends on half a dozen new extraordinary results in Applications and Compu- tational Mathematics and it shortens their CPU run times by up to 85 %. A new class of Threshold based Matrix Algorithms helps us along here. These new Results from the early 2020s are all in- terconnected, some of these date back a hundred or more years. They all hinge on new branches and new parts of Matrix Theory and Matrix Computa- tions. These were developed by the author, with help from many others, over many decades, along the way. Our intended aim all along was to discover block- decomposable matrices and matrix flows as they might originate in real-time from time-varying sensor data. For block-diagonalisable nby nAor A(t), the complexity of their numerical treatment decreases swiftly for O(n3) matrix processes when adopting ‘divide and conquer’ methods on each of their diagonal blocks separately, see the Appendices in [56].