More Alain Connes....

 

 

 https://arxiv.org/pdf/math/0404128.pdf

 

 So now let's go back to the music...

 

 

arXiv:hep-th/0411114  [pdf, ps, other] 

hep-th math-ph math.AG math.NT math.QA

From Physics to Number Theory via Noncommutative Geometry, Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory

Authors: Alain Connes, Matilde Marcolli

Abstract: We establish a precise relation between Galois theory in its motivic form with the mathematical theory of perturbative renormalization (in the minimal subtraction scheme with dimensional regularization). We identify, through a Riemann-Hilbert correspondence based on the Birkhoff decomposition and the t'Hooft relations, a universal symmetry group (the "cosmic Galois group" suggested by Cartier),… ▽ More

Submitted 11 November, 2004; originally announced November 2004.

 eigenvalue 1λ

https://www.quora.com/What-is-the-relation-of-a-matrix-inversion-and-eigenvalues

So  Connes is using Inverse Frequency as the eigenvalue.

 in general, if the eigenvalues of an n×n invertible matrix, M, are λ1,λ2,…,λn, then the eigenvalues of M−1 are 1λ1,1λ2,…,1λn.

 

Which we can rearrange as:

1λv=M−1v

Which implies that v

is also an eigenvector of M−1 with eigenvalue 1λ.

 So the inverse frequency is actually the invertible Matrix that is noncommutative as the inner products of the Matrix. Fascinating!!

If we take the canonical definition of eigenvectors and eigenvalues for a matrix, M, and further assume that M is invertible, so there exists, M−1 such that MM−1=M−1M=I

, then we can see that:

Mv=λv

Multiply both sides by M−1

:

M−1Mv=λM−1v

So:

v=λM−1v

Put another way, a matrix and it's inverse share eigenvectors, but their eigenvalue are inverses of each other.