Keywords: math history, numerical analysis, fractional ODEs, shooting methods, proportional secting, Linear algebra, unitary block-decomposition, invariant subspace theory, quantum physics, time-varying matrix problem, conditionally stable algorithm, Johnson matrix flow, neural network, zeroing neural network, discretized ZNN algorithm, field of values, matrix flows, time-varying numerical algorithm, ZNN set-up, predictive numerical method, matrix symmetrizer

AMS : 65F10, 65F15, 65F30, 65J10, 65L10, 34A08, 35A08, 15A18, 15A21, 15A23, 15A60, 15B57, 81Q10

An Introduction – of sorts

For general complex or real 1-parameter matrix flows \(A(t)_{n,n}\) or for static matrices \(A\)this paper considers ways to decompose matrix flows \(A(t)\) or single matrices \(A_{n,n}\) globally via one constant hermitean matrix similarity \(C_{n,n}\) as

\[A(t) = C^{-1} \cdot diag(A_1(t), ..., A_\ell (t)) \cdot C\]

or

\[A = C^{-1}\cdot diag(A_1, ..., A_\ell ) \cdot C .\]

Here each diagonal block \(A_k(t)\) or \(A_k\) is square and their number \(\ell \) exceeds 1 – if this is possible.

The theory behind our proposed algorithm is elementary and uses the concept of invariant subspaces for the MATLAB eig computed ‘eigenvectors’ of another associated flow matrix \(B(t_a)\) to find the coarsest simultaneous block structure for all flow matrices \(B(t_b)\) and consequently block-diagonalizes the given matrix flow \(A(t)\) or the given static matrix \(A\) itself.

The method works in \(O(n^2)\) time for all matrices \(A_{n,n}\) and all matrix flows \(A(t)\), be they real or complex, normal, with Jordan structures or repeated eigenvalues, and differentiable, continuous, or discontinuous.

We aim to discover unitarily diagonal-block decomposable matrices and flows from sensor given data. For unitarily block-diagonalisable \(A\) or \(A(t)\), the complexity of their numerical treatment decreases swiftly for \(O(n^3)\) matrix processes when working on each of their diagonal blocks separately.

The proof : The Unitary Block-Decompositions and
Fast and Accurate Field of Values Boundary Computations
of all Square Matrices

The unitary block decomposition of matrix flows or single square matrices has never been resolved by algebraic means in 100 + years. Our new way of testing and establishing unitary decomposability of a matrix flow or a static matrix hinges on a numerical algorithm that decides the lay of the near zero entries and of the sizable ones in the hermitean Johnson matrix flow

\[ \hspace *{24mm} {\mathbf { F(t) = cos(t)H + sin(t)K}} \]

for \(0 \leq t \leq 2 \pi \).

[ We assume today that all matrices and matrix flows are real in this talk. ]

The algorithm starts from two linear independent Johnson flow matrices \(\mathbf {F}(t_1)\) and \(\mathbf {F}(t_2)\). We form the 0-1 logic matrices \(Flogic(t_1)\) and \(Flogic(t_2)\) in Matlab’s spy function and assemble the normalized eigenvectors of \(\mathbf {F}(t_1)\) in \(V'(t_1)\) so that adjacent rows in \(Flogic(t_1)\) have equal 0 and 1 patterned blocks.

We reorder the rows of \(Flogic(t_2)\) so that they have the same 0-1 pattern as \(Flogic(t_1)\) by using a combinatorial algorithm that establishes a joint proper unitary block decomposition of both logic 0-1 spy pattern matrices.

Going down from the top block leads to a joint unitary block diagonal structure for all \(F(t)\) with \(t \in [0, 2\pi ]\). Since \(F(0) = H\) and \(F(\pi /2) = K\), the matrix \(A\) or the flow \(A(t)\) have the same unitary block structure as the hermitean matrices \(F(t_1)\) and \(F(t_2)\) do, and our algorithm is complete.

This algorithm depends on the magnitude relations between the non-zero entries and the near zero entries in \(F(t_i)\). To distinguish between computed entries as being ’zero’ or definitely ’nonzero’ is a an uncharted question. We have set the ’zero’ threshold heuristically to \(\| A \|_{fro} \cdot 10^{-13} \approx \| A \|_{fro} \cdot eps\) when working in double precision in Matlab.


To prove the main Unitary Diagonability Theorem, we assume that the hermitean Johnson flow \(F (t)\) contains two linearly independent matrices \(F(t_a ) \neq F(t_b )\) where \(F (t) = \cos (t)H + \sin (t)K\) for the hermitean and skew parts \(H = (A+A^*)/2 = H^*\) and \(K = (A-A^*)/(2i) = K^*\) of \(A\).

We know of no way to prove the unitary block-diagonalization theorem algebraically; an algebraic proof of our result was never attempted. An algorithmic proof was nearly impossible before the advent of computers and the understanding of matrix flows such as the Johnson flow \(\mathcal {F}(t)\) and of the predictive qualities and accuracy of Zhang Neural Networks.

The ability to construct the field of values \(F (A)\) boundary curve of some unitarily decomposable matrices quickly and accurately via shooting methods led to recent further studies of matrix eigencurve behavior by many who tried to locate their crossings or hyperbolic avoidances that had first appeared in original Quantum Physics studies in Copenhagen, Berlin and Göttingen in the 1920s. Albert Einstein shunned this fundamental Quantum Theory problem and rather worked on relativity. Bohr’s group wanted to understand eigencurve crossings and their relation to the unitary matrix block-decomposability of matrices, but neither succeeded.

Eigencurve crossing studies for the unsolved unitary block-diagonalization problem began anew with Charlie Johnson’s work on the field of values in the 1970s and advanced in the 1990s in Luca Dieci et al’s, Loisel and Maxwell’s, and my eigencurve papers. By 2020 several small sub-results of the unitary block-decomposition problem had been solved, but no complete classification had been found and the century old problem was still open in 2023.

Our final classification of unitary block-decompositions for both, general matrix flows and static matrices, was inspired by Yunong Zhang’s parameter-varying Neural Networks that date back to his doctoral thesis of 2001.

These recent developments are inter-woven and all have contributed to each other’s solution and to the author’s understanding of this cluster’s new fundamental computational results. Each of the new results has advanced its area tremendously where no math methods had been found for decades, by researchers that relied on standard mathematical or computational approaches.

The mathematical process and its results were never predictable – until everything fell into place in 2023/24, was mathematically sound, proved, and their papers quickly accepted.

The conditionally stable Matrix Computational algorithms of this research cluster deliver highly accurate results at high speed and they do not suffer from the inaccuracies of our the traditional ’backward stable’ methods

This is the end of my math talk with selections from the new extraordinary mathematical research cluster.

What is our most urgent task today, in Linear Algebra and Matrix Theory, teaching and research wise ?
What needs to be done immediately ?

We must modernize our early College teachings in Linear Algebra and enliven this area of Mathematics that helps us all so extraordinarily in our daily cell-phone and internet based lives and in our AI and engineering advances.

How do we, how do you mostly teach beginners’ linear algebra

classes today?

Which subjects do we, you, and I teach?

Using modern Matrix Theory based ideas or from a classical and algebraic standpoint ?

Top-down taught or taught interactively ?   That is the question

Math Education : the Necessity of Modernizing the First Linear Algebra Course via coherent Matrix Theory Based Lesson Plans

This is the subject of a serious Math Education session, some time late at night, in Ithaka. All are welcome