A synchronized dance of eigenvalues

September 26, 2017*

*Last modified 11-Nov-19

Tags: math, visualization, numerics

Gif of four dots moving inside the unit circle. Gif of four dots moving inside the unit circle. Gif of four dots moving inside the unit circle. Gif of four dots moving inside the unit circle. Gif of four dots moving inside the unit circle. Gif of four dots moving inside the unit circle. Gif of four dots moving inside the unit circle. Gif of four dots moving inside the unit circle. Gif of four dots moving inside the unit circle.
Gif of four dots moving inside the unit circle. Gif of four dots moving inside the unit circle.

The motivation behind this post is to show some off some nice gifs. But I thought maybe they aren’t actually interesting without any context, so below I’ll try to explain what the gifs are about.

Brief mathematical introduction to Perron-Frobenius theory

Consider the set of n×nn\times n matrices MnM_n, with complex entries. The algebraic structure (scalar multiplication by complex numbers, addition and multiplication of matrices) plays well with the operator norm, ∥A∥=sup⁡v∈Cn:∥v∥2=1∥Av∥2 \|A\| = \sup_{ v\in \mathbb{C}^n: \|v\|_2=1 } \|A v\|_2 and the involution ∗* (i.e. conjugate-transpose), making it a C∗^*-algebraI recommend p. 17 of Jane Panangaden’s thesis for an excellent diagram explaining the interplay between these structures, along with much more on C∗^*-algebras. Panangaden, J. (2016), Energy Full Counting Statistics in Return-to-Equilibrium. McGill University, Montreal QC.

. A matrix A∈MnA\in M_n is positive semi-definite if it can be written as A=B∗BA= B^*B for some other matrix B∈MnB\in M_n.

Let us now go one level higher: we will consider linear maps ϕ\phi from MnM_n to MnM_n, i.e. linear maps whose arguments are themselves matrices. Such a linear map is positive if it maps positive elements of MnM_n to positive elements of MnM_n. Since ϕ\phi is a linear map on a vector space (even if that vector space is itself the set of matrices), it has eigenvalues and eigenvectors. The eigenvalues are in general complex numbers, and number r=max⁡{∣λ∣:λ is an eigenvalue of ϕ} r = \max\{ |\lambda| : \lambda \text{ is an eigenvalue of }\phi \} is the spectral radius of ϕ\phi.

With these definitions, we can consider the following Perron-Frobenius type theorem considered by Evans and Høegh-Krohn in 1978Evans, D. E. and Høegh-Krohn, R. (1978), Spectral Properties of Positive Maps on C∗^*-Algebras. Journal of the London Mathematical Society, s2-17: 345–355.

:

Theorem 1 Let ϕ\phi be a positive linear map on MnM_n. If rr is the spectral radius of ϕ\phi, then there is a non-zero positive element A0∈MnA_0\in M_n such that ϕ(A0)=rA0. \phi( A_0) = r A_0.

This may sound abstract (or maybe not), but it is quite nice. A priori, we don’t know that the spectral radius itself is an eigenvalue; from the definition, we only know there is an eigenvalue whose absolute value is the spectral radius. But the above theorem tells us that if ϕ\phi is a positive map, then indeed the spectral radius itself is an eigenvalue, and that the corresponding eigenvector A0A_0 is positive.

This result is relevant, e.g., for understanding repeated interaction systems. These are discrete-time quantum systems, and the time evolution is given by a positive linear map with spectral radius 1. The quantum states ρ\rho are given by positive elements of MnM_n with trace 1. The above theorem then tells us that there is automatically an invariant state, i.e. a quantum state ρ\rho such that ϕ(ρ)=ρ\phi(\rho)=\rho. Note we are using both that the spectral radius is an eigenvalue (so there is an invariant matrix), and that the corresponding eigenvector is positive (so once we divide by its trace, it is a quantum state).

Let us assume A0A_0 is invertible. We can define a modified version of the map ϕ\phi by ϕ^(B)=1rA0−1/2ϕ(A01/2BA01/2)A0−1/2∈Mn(1) \hat \phi (B) = \frac{1}{r} A_0^{-1/2} \phi(A_0^{1/2} B A_0^{1/2} )A_0^{-1/2} \in M_n \qquad(1) for B∈MnB\in M_n. In fact, this map comes up in the proof of the theorem. Now, with the knowledge of the theorem, we see that this map acts on the identity matrix II by ϕ^(I)=1rA0−1/2ϕ(A01/2A01/2)A0−1/2=1rA0−1/2(rA0)A0−1/2=I \hat \phi (I) = \frac{1}{r} A_0^{-1/2} \phi(A_0^{1/2} A_0^{1/2} )A_0^{-1/2} = \frac{1}{r} A_0^{-1/2} (r A_0) A_0^{-1/2} = I so in fact, ϕ^\hat \phi has the eigenvalue 1 with associated eigenvector II.

A particular deformation

A particular typeThese are the completely positive maps, which are usually defined in a different way, but the usual definition is equivalent to the one I’ll give here.

of positive linear map on MnM_n is one which can be represented as ϕ(B)=∑i=1kVi∗BVi \phi (B) = \sum_{i=1}^{k} V_i^* B V_i for some V1,…,Vk∈MnV_1,\dotsc,V_k \in M_n. The matrices ViV_i are called the Kraus operators. So ϕ\phi acts on BB by conjugating it by each ViV_i and adding up the result.

Let us fix such a map ϕ\phi, and moreover assume that ϕ\phi has spectral radius 1, with the associated eigenvalue as the identity matrix. By the definition, we see this implies ϕ(I)=∑i=1kVi∗Vi=I. \phi (I) = \sum_{i=1}^{k} V_i^* V_i = I.

It turns out when considering the statistics of measurements of a repeated interaction systemin which case, the ϕ\phi of this section is actually the adjoint of the time-evolution operator

, a particular deformation of ϕ\phi becomes important. Let v∈Rkv\in \mathbb{R}^k be a real vector with strictly positive entries. Then define ϕ(α)(B)=∑i=1kviαVi∗BVi \phi^{(\alpha)} (B) = \sum_{i=1}^{k} v_i^\alpha V_i^* B V_i for α∈R\alpha\in \mathbb{R}. Note that ϕ(0)=ϕ\phi^{(0)} = \phi. For real α\alpha this map is still positivein fact, still completely positive, with Kraus operators viα/2Viv_i^{\alpha/2} V_i.

, but no longer has spectral radius 1 or ϕ(α)(I)=I\phi^{(\alpha)}(I) = I, in general. However, by the theorem above, the spectral radius r(α)r(\alpha) is an eigenvalue of ϕ(α)\phi^{(\alpha)}, and has a corresponding eigenvector A0(α)A_0(\alpha) which is a positive element of MnM_n. Let us consider the modification of the map given in (eq. 1), defined by ϕ^(α)(B)=1r(α)A0(α)−1/2ϕ(α)(A0(α)1/2BA0(α)1/2)A0(α)−1/2∈Mn \hat \phi^{(\alpha)} (B) = \frac{1}{r(\alpha)} A_0(\alpha)^{-1/2} \phi^{(\alpha)}(A_0(\alpha)^{1/2} B A_0(\alpha)^{1/2} )A_0(\alpha)^{-1/2} \in M_n for B∈MnB\in M_n. Now, we automatically have ϕ^(α)(I)=I\hat \phi^{(\alpha)} (I) = I. So the undeformed map, ϕ\phi has this property, and so does ϕ^(α)\hat \phi^{(\alpha)}. That is, we know what happens to the eigenvalue 1 and the associated eigenvector when we go from ϕ\phi to ϕ^(α)\hat \phi^{(\alpha)}: they stay the same, for any α\alpha.

But what happens to the other eigenvalues?

That’s the motivating question for these numerics. Maybe they don’t change either. Maybe they all shrink to zero as α\alpha increases. Maybe they just rotate around in a circle. To investigate this, I randomly generated some maps ϕ\phi, and looked what happened to the eigenvalues of ϕ^(α)\hat \phi^{(\alpha)} while changing α\alpha in the interval [0,1][0,1]. And of course, the resulting animations are the gifs at the top of the post. The last two are examples of particular interest, and are not randomly generated– and nothing movesActually, shortly after I posted this I realized that in the second example, the eigenvalues do move— just very slowly, and it’s not really visible in this gif.

! But for the general (random) ones, it looks like almost anything can happen. Cool!

More information

The preprint arxiv/1705.08281 has much more motivation for why we are interested in the maps ϕ(α)\phi^{(\alpha)} and ϕ^(α)\hat \phi^{(\alpha)}. In particular, ϕ(α)\phi^{(\alpha)} is the (Hilbert-Schmidt) adjoint of the map LY(α)(s)\mathcal{L}_Y^{(\alpha)}(s) which is featured in Proposition 3.5. The Perron-Frobenius theory of these deformed maps is investigated in Appendix A. The two examples for which none of the eigenvalues move with α\alpha are Example 4.5 and Example 5.11 in the preprint.

The numerics were done in Julia, and the raw code is here. I also have it as an IJulia notebook available here. And if you just want to scroll through it, an HMTL formatted version is here.

More eigenvalues

Gif of four dots moving inside the unit circle. Gif of four dots moving inside the unit circle. Gif of four dots moving inside the unit circle. Gif of four dots moving inside the unit circle. Gif of four dots moving inside the unit circle. Gif of four dots moving inside the unit circle. Gif of four dots moving inside the unit circle. Gif of four dots moving inside the unit circle. Gif of four dots moving inside the unit circle.