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June 2018, Bohemian Workshop
On the number of different real eigenvalues
of NxN Upper Hessenberg Toeplitz
Bohemian matrices when N is fixed
Laureano Gonzalez-Vega Universidad de Cantabria, Spain
together with Rob, Eunice, Steven, Juany and Rafa
June 2018, Bohemian Workshop
Problemathand
Data:Popula)on:-1,0and1.Type ofmatrices: Upper Hessenberg and Toeplitzmatricesnxnwith1inthefirstsubdiagonal.SetdenotedbyUHTn.Numberofelements:3n.
Output:Foreveryn,computethedistribu)onofthenumberofdifferentrealeigenvalues.
June 2018, Bohemian Workshop
Problemathand
UHTn
June 2018, Bohemian Workshop
Problemathand
UHTn
June 2018, Bohemian Workshop
Problemathand
UHTn
June 2018, Bohemian Workshop
n=3:Thereareonly27UpperHessenbergandToeplitzmatrices3x3with-1inthefirstsubdiagonaland-1,0and1entries.
Output:0differenteigenvalues:0matrices.1differenteigenvalues:18matrices.2differenteigenvalues:0matrices.3differenteigenvalues:9matrices.
Example
June 2018, Bohemian Workshop
Example
June 2018, Bohemian Workshop
Maplebased:Easygenera)onofUHTn.Characteris)cpolynomials(integercoefficients).Numberof different real roots computedbyusingSturm’ssequences
Experiments
June 2018, Bohemian Workshop
n 0 1 2 3 4 5 6 7 8 9 10
3 0 18 0 9
4 21 6 39 9 6
5 0 132 6 87 6 12
6 177 12 324 9 195 6 6
7 0 999 18 777 6 378 0 9
8 1395 27 2682 84 1800 36 525 6 6
9 0 7326 78 6939 54 4398 36 843 0 9
10 11244 117 20133 333 16674 198 9480 78 783 3 6
Output
June 2018, Bohemian Workshop
n 0 1 2 3 4 5 6 7 8 9 10
3 0 18 0 9
4 21 6 39 9 6
5 0 132 6 87 6 12
6 177 12 324 9 195 6 6
7 0 999 18 777 6 378 0 9
8 1395 27 2682 84 1800 36 525 6 6
9 0 7326 78 6939 54 4398 36 843 0 9
10 11244 117 20133 333 16674 198 9480 78 783 3 6
Output
June 2018, Bohemian Workshop
Letnbefixed.Let rk the number of matrices in UHTn with exactly kdifferentrealeigenvalues.Then:
Ifnisoddthenr1>rkforanyk≠1.Ifniseventhenr2>rkforanyk≠2.
Theconjecture
June 2018, Bohemian Workshop
Spectral Properties of Banded Toeplitz Matrices Albrecht Böttcher and Sergei M. Grudsky SIAM 2005 This chapter is devoted to the results of Schmidt, Spitzer, and Hirschman on the asymptotic eigenvalue distribution of Tn(b) as n → ∞. We show that the spectra sp Tn(b) converge in the Hausdorff metric to a certain limiting set Λ(b), which is either a singleton or the union of finitely many analytic arcs.
Asymptotics of eigenvalues and eigenvectors of Toeplitz matrices Hui Dai, Zachary Geary and Leo P KadanoffJournal of Statistical Mechanics: Theory and Experiment, May 2009
Theconjecture(andtheexclusionarea)
June 2018, Bohemian Workshop
On the number of Neutral Matrices in the set of NxN
Upper Hessenberg Toeplitz Bohemian matrices
when N is fixed
June 2018, Bohemian Workshop
Problemathand
Data:Popula)on:-1,0and1.Type ofmatrices: Upper Hessenberg and Toeplitzmatricesnxnwith1inthefirstsubdiagonal.SetdenotedbyUHTn.Numberofelements:3n.
Output:Foreveryn,computethenumberofneutralmatricesinUHTn.
Amatrixisneutralifalleigenvaluesareintheimaginaryaxis.
June 2018, Bohemian Workshop
Maplebased:Easygenera)onofUHTn.Characteris)cpolynomials(integercoefficients).
Experiments:juststarCng
P(x) = det(x𝕀n − A) =n
∑j=0
ajxj
Aisneutralifandonlyif:
P(x) = xs(x2m + a2m−2x2m−2 + … + an−2m)
aj ≥ 0, ∀j
tm + a2m−2tm−1 + … + an−2m has only real roots
June 2018, Bohemian Workshop
On the number of Correlation Matrices
in the set of NxN Bohemian matrices
when N is fixed
June 2018, Bohemian Workshop
Problemathand
Data:Popula)on:0and1or-1,0and1.Typeofmatrices:nxnBohemianMatrices.BM[0,1]orBM[-1,0,1].
Output:For every n, compute the number of correla)on matrices inBM[0,1]orBM[-1,0,1].
Nick
June 2018, Bohemian Workshop
Problemathand
Nick
June 2018, Bohemian Workshop
Problemathand
All eigenvalues of A are nonnegativeif and only if
A is positive semidefiniteif and only if
Alternating signs for the coefficients of the characteristic polynomial of A
June 2018, Bohemian Workshop
Example:n=3
June 2018, Bohemian Workshop
Example:n=3
June 2018, Bohemian Workshop
Example:n=3
Correlation matrices. Characterisation when n=3:
June 2018, Bohemian Workshop
Example:n=3
Correlation matrices. Characterisation when n=3:
P. J. Rousseeuw and G. Molenberghs: The Shape of Correlation Matrices. The American Statistician 48, 276-279, 1994.
Nick: Thanks !
June 2018, Bohemian Workshop
Example:n=3
BM[0,1]
June 2018, Bohemian Workshop
Example:n=3
BM[-1,0,1]
June 2018, Bohemian Workshop
Example:n=4
Correlation matrices. Characterisation when n=4:
June 2018, Bohemian Workshop
Maplebased:FirstcasenotinNick’stable:n=8.Coefficients of the characteris)c polynomial areeasytocomputeandevaluate.
Experiments
June 2018, Bohemian Workshop
ExperimentsCorrelation matrices. Characterisation when n=4:
What about the geometry of this semialgebraic set in ?ℝ6
June 2018, Bohemian Workshop
Too many interesting things to do !
Conclusions
