2020/02/04 by Frank Uhlig, Uhlig, Frank
Computer Science · Mathematics · #15A18 #15A60 #65F15 #65F30 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Combinatorics #Conjecture #Eigenvalues and eigenvectors #FOS: Mathematics #Jordan matrix #Mathematical analysis #Mathematics #Matrix (chemical analysis) #Normal matrix #Numerical Analysis (math.NA) #Physics #Pure mathematics #Quantum mechanics #Tensor decomposition and applications #Unitary matrix #Unitary state #cs.NA #math.NA #msc:15A18 #msc:15A60 #msc:65F15 #msc:65F30
paper · pdf · doi:10.48550/arxiv.2002.01274
15 pages, 11 graphs
arxiv created 2020/02/04 · openalex publication_date 2020/02/04 · arxiv updated 2020/02/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We investigate the eigenvalue curves of 1-parameter hermitean and general\ncomplex or real matrix flows A(t) in light of their geometry and the uniform\ndecomposability of A(t) for all parameters t. The often misquoted and\nmisapplied results by Hund and von Neumann and by Wigner for eigencurve\ncrossings from the late 1920s are clarified for hermitean matrix flows A(t) =\n(A(t))^*. A conjecture on extending these results to general non-normal or\nnon-hermitean 1-parameter matrix flows is formulated and investigated. An\nalgorithm to compute the block dimensions of uniformly decomposable hermitean\nmatrix flows is described and tested. The algorithm uses the ZNN method to\ncompute the time-varying matrix eigenvalue curves of A(t) for to \≤ t\≤\ntf. Similar efforts for general complex matrix flows are described. This\nextension leads to many new and open problems. Specifically, we point to the\ndifficult relationship between the geometry of eigencurves for general complex\nmatrix flows A(t) and a general flow's decomposability into blockdiagonal\nform via one fixed unitary or general matrix similarity for all parameters t.\n