Calculus without limits

# Kublanovskaya-Francis Transform of Dirac matrix

Since finding the isospectral deformation of the exterior derivative (see “An integrable evolution equation in geometry” from June 1, 2013 and “Isospectral Deformations of the Dirac operator” from June 24, 2013), I tried to find discrete time integrable evolutions of the Dirac operator. Last Sunday, while experimenting in a coffee shop, I dared to look at what I should have tried a long time ago: Take the Dirac matrix D and apply the QR algorithm on it. This produces a new isospectral matrix. Once you do that it , it becomes obvious that the new matrix D is again a Dirac matrix and so defines an isospectral equivalent exterior derivative on the simplicial complex by looking at the new Dirac matrix D’= RQ which is isospectral because Q D’ Q* = D! The proof is obvious once one realizes that the orthogonal Q matrix has the same Dirac form structure as D. Note that since L is selfadjoint, the spectral theorem gives an orthgonal Q which diagonalizes L. My observation is that we can do that so that still, the diagonal L comes from an exterior derivative d: we have $L=(d+d^*)^2$ with a derivative meaning $d^2=0$ and that $d$ maps k-forms to k+1 forms. As for the Dirac matrix, see this talk at ILAS in 2013.

Observation (July 28, 2024): If a simplicial complex has a Laplacian $L= (d+d*)^2$ for which all eigenvalues are different, then there exists an equivalent exterior derivative d’ obtained by applying the QR flow.

The QR algorithm (Kublanovskaya-Francis transformation from 1961) produces so new exterior derivative matrices d’ (incidence matrices considered by Poincare already) with algebraic numbers as entries for which $L'=(d'+d'^*)^2$ is arbitrary close to a diagonal matrix and again has matrix entries containing algebraic numbers. The statement about algebraic numbers is relevant because we have so not only numerical but exact expressions for the exterior derivative. The observation is rather obvious, once one has seen it.

We should add that the discrete isospectral deformation of the Dirac matrix is of a different nature like the Toda-Lax type deformation which I had found in 2013. In that case, the deformed Dirac operator gained a diagonal part $D=d+d*+b$. The new Dirac operator $d+d^*$ then changes the Connes distance in space. It expands space and this is quite universal. It does not matter, whether we take a finite setting like a finite abstract simplicial complex, or the frame work in a smooth compact Riemannian manifold. Space expands with an inflationary start. In the QR case, the Connes distance is preserved. It is a discrete symmetry of space. It can be interpolated of course by a quantum mechanical (Heisenberg picture) way. In the finite case it follows from the fact that SU(n) is connected and any Q can be written as Q=exp(i A) with symmetric A and so interpolated by the Schroedinger equation u’=i A u.

Maybe a bit of history first: the numerical diagonalization method of QR was developed around 1958-1960 independently by John Francis and Vera Kublanovkaya (born 1920) . They were both motivated by the LR algorithm of Rutishauser who wrote a paper about this in 1958 but had relations with earlier work of Eduard Stiefel and Rutishauser on Quotienten-Differenzen algorithms. Francis reportedly was also motivated by work of Olga Taussky Todd done during 1940ies in the context of stability of planes. See the 2009 paper of Golub and Uhlig on the QR algorithm. Vera Kublanovskaya was alive still in 2008, when the review of Golub and Uhlig were written. She passed away in 2012. John G.F. Francis is an English mathematician. See the Wikipedia entry.

Maybe a bit of more personal (and ETH) connections. Heinz Rutishauser (1918-1970) was a Swiss pioneer in computer science. He had visited in 1948/1949 Harvard University. At that time, the Mark I computer was still operative. The machine had been in the Science center until 2021. Besides Rutishauser, also Eduard Stiefel (1909-1978) was an interesting figure in computer science and mathematics. He was trained as a pure mathematician (his advisor was Heinz Hopf, who was of huge influence to ETH. As I pointed out a couple of times already, half of my ETH teachers, including my high school teacher Roland Staerk were descendants (academic kids or grand-kids) of Heinz Hopf. As Olga Taussky Todd (the torch bearer of matrix theory) was mentioned above in the context of the QR history, I should add again that I was lucky to meet her in Pasadena (winter 2014) shortly before her passing in 2015. She had wanted to meet the Olga-Taussky-Todd instructors at Caltech (I was one) and came once to the Caltech math department.

An other a bit more personal connection to QR is my PhD thesis. Much of my thesis is about isospectral deformations of random operators.

[I should maybe caution again here that random operators are understood in the sense that they are operator valued random variables, there does not need anything to be “random” in a colloquial sense about it. The almost Matthieux operator for example is an operator valued random variable, where the probability space is the circle and where the entries are correlated in an almost periodic way. In probability theory, any function $X: \Omega \to \mathbb{R}$ is called a random variable and also there, there is nothing a priory which asks that a random variable to anything random. Randomness comes only in if one looks at processes. Part of probability theory deals so with IID (independent, identically distributed) random variables. In that case, randomness is more like what we intuitively mean when we talk about “random”. For example, every measurable function on the unit interval [a,b] is a random variable when we consider the probability space $([a,b],\mathcal{A},dx/(b-a))$. Especially, every continuous function f on an interval, an object in calculus also can be seen as a random variable. As Joseph Doob once pointed out, any sequence of random variables $X_k$ (a discrete time stochastic process) which have the same distribution can be seen as coming from a probasbility space $(\Omega, \mathcal{A},P)$ , a single random variable X and a transformation $T: \Omega \to \Omega$. The stochastic process is then $X_k(x) = X(T^kx)$. In some sense, probability theory is part of dynamical systems theory, similarly than statistics is part of mathematics! ]

My graduate student work had been motivated by the unsolved problem to find the entropy of the Standard map. It is the problem to compute the Kolmogorov-Sinai entropy log(det(L)) for a random operator. The operator L the Hessian of the underlying variational principle, defining the dynamics. An isospectral deformation of L of course preserves the value log(det(L)) and I had also been looking for ways to deform L to get more insight. I also had considered the QR flow, which has relations to Backlund transformations. There is an important underlying principle for integrable systems that any integrable system seems to be in some way is related to isospectral deformation and so also to algebraic geometry. l had tried to push the integration of the periodic Toda lattice (which by Abelian integrals becomes a linear flow on a torus) to the random case. In the periodic Jacobi case, there is a nice hyperelliptic curve defined and the algebro-geometric frame work works. In some sense, what I’m working on now (Dirac matrices for arbitrary simplicial complexes or delta sets) is very much related to the work I had been doing then. At that time, I wrote L-E as a square of a Dirac operator (again a Jacobi matrix) but on a doubled lattice. This implements the inverse of the quadratic map on a space of Jacobi matrices and the attractor becomes a “quantum Julia set”. The spectra of the points in this quantum Julia set are operators and the spectrum of the operator associated to the energy E is the classical Julia set of the quadratic map $f(z) = z^2+E$.

I just animated the two pioneers about the QR algorithm: Vera Nikolaevna Kublanovskaya (1920-2012) and John G.F. Francis (1934- ) (I hope he does not mind being animated. I usually only animate historical figures.) Both were before 2008 (when the Golub-Uhlig article came out) pretty unknown. Despite the fact that QR algorithm is considered one of the 10 most important algorithms in all of computer science.