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BOHEMIAN MATRICES: MISE EN PLACE
Juana Sendra PonsEscuela Técnica Superior de Ingenieria y Sistemas de
Telecomunicación.Universidad Politécnica de Madrid
21 de junio de 2018
Laureano González Vega
BOHEMIAN MATRICES
ROBERT M. CORLESS
Juana Sendra Pons
Rafael Sendra Pons
Universidad de Cantabria
Universidad de Alcalá
Universidad Politécnica de Madrid
Department of Applied Mathematics Western University
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 2 / 58
Laureano González Vega
BOHEMIAN MATRICES
Department of Applied MathematicsWestern University Ontario
ROBERT M. CORLESS
Juana Sendra Pons
Rafael Sendra Pons
Universidad de Cantabria
Universidad de Alcalá
Universidad Politécnica de Madrid
Eunice Y. S. Chan
Steven Thornton
Department of Applied Mathematics Western University
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 3 / 58
Laureano González Vega
BOHEMIAN MATRICES
Department of Applied MathematicsWestern University
ROBERT M. CORLESS
Juana Sendra Pons
Rafael Sendra Pons
Universidad de Cantabria
Universidad de Alcalá
Universidad Politécnica de Madrid
Eunice Y. S. Chan
Steven Thornton
Department of Applied Mathematics Western University
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 4 / 58
Laureano González Vega
BOHEMIAN MATRICES
Department of Applied MathematicsWestern University
ROBERT M. CORLESS
Juana Sendra Pons
Rafael Sendra Pons
Universidad de Cantabria
Universidad de Alcalá
Universidad Politécnica de Madrid
Eunice Y. S. Chan
Steven Thornton
Department of Applied Mathematics Western University
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 5 / 58
Mise en place
In Gourmet cooking "Mise en place"means the task of organizingand arranging the ingredients before cooking.
The mise en place for the spanish paella.............
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 6 / 58
Mise en place
In Gourmet cooking "Mise en place"means the task of organizingand arranging the ingredients before cooking.The mise en place for the spanish paella.............
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 6 / 58
Mise en place
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 7 / 58
I refer to the Mise en place of Bohemian Matrices, as the task ofarranging the ingredients (experiments and conjectures) before cookthe theory.
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 8 / 58
New Challenges
We are interested in analyzing the behaviour of the eigenvalues ofthe Bohemian Matrices.
Are the eigenvalues real?Are the eigenvalues simple?How do these eigenvalues behave?How many different characteristic polynomials are there?How many of these characteristic polynomials are square–free?Which are the infinity norms of these characteristicpolynomials?How many Rhapsodic matrices are there?..............................
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 9 / 58
New Challenges
We are interested in analyzing the behaviour of the eigenvalues ofthe Bohemian Matrices.
Are the eigenvalues real?Are the eigenvalues simple?How do these eigenvalues behave?How many different characteristic polynomials are there?How many of these characteristic polynomials are square–free?Which are the infinity norms of these characteristicpolynomials?How many Rhapsodic matrices are there?..............................
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 9 / 58
Several different experiments:
Upper Hessenberg Toeplitz Bohemian matricesSquare–free characteristic polynomials,polynomials with real roots,Rhapsodic matrices.......
Toeplitz Bohemian matricesSquare–free characteristic polynomials,polynomials with real roots,Rhapsodic matrices.......
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 10 / 58
Several different experiments:
Upper Hessenberg Toeplitz Bohemian matricesSquare–free characteristic polynomials,polynomials with real roots,Rhapsodic matrices.......
Toeplitz Bohemian matricesSquare–free characteristic polynomials,polynomials with real roots,Rhapsodic matrices.......
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 10 / 58
Upper Hessenberg Toeplitz Bohemian matrices
n Nº of matrix
Nº of characteristicpolynomial
Nº sqr-free characteristicpolynomial
Nº poly with all real roots
Nº poly with n simple real roots
Nº poly with non real roots
Nª Rhapsodic
2 9 9 6 6 3 3 4
3 27 27 24 12 9 0 10
4 81 81 66 18 6 21 16
5 243 243 228 21 12 0 42
6 729 729 684 21 6 177 62
7 2187 2187 2142 21 9 0 134
8 6561 6561 6382 24 6 1395 188
9 19683 19683 19458 21 9 0 382
10 59049 59049 58230 21 6 11244 532
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 11 / 58
Upper Hessenberg Toeplitz Bohemian matrices
n Nº of matrix
Nº of characteristicpolynomial
Nº sqr-free characteristicpolynomial
Nº poly with all real roots
Nº poly with n simple real roots
Nº poly with non real roots
Nª Rhapsodic
%
2 9 9 6 6 3 3 4 44,4%
3 27 27 24 12 9 0 10 37,03%
4 81 81 66 18 6 21 16 19,75%
5 243 243 228 21 12 0 42 17,28%
6 729 729 684 21 6 177 62 8,5%
7 2187 2187 2142 21 9 0 134 6,12%
8 6561 6561 6382 24 6 1395 188 2,8%
9 19683 19683 19458 21 9 0 382 1,9%
10 59049 59049 58230 21 6 11244 532 0,9%
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 12 / 58
New Challenges
Rhapsodic matrices
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 13 / 58
New Challenges
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 14 / 58
New Challenges: Toeplitz Bohemian matrices
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 15 / 58
New Challenges
n Nº of matrix
Nº of charact poly
Nº of square-free chpoly
Nº poly with all real roots
Nº poly with non real roots
Nº Rhapsodic
3 9 6 3 0 0 1
4 27 11 3 0 0 15 81 24 18 0 0 76 243 70 25 0 0 17 729 130 116 0 0 258 2187 341 162 0 0 19 6561 1104 843 0 0 37
10 19683 2380 1637 0 0 711 59049 5934 5880 0 0 121
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 16 / 58
New Challenges
n Nº of matrix
Nº of charact poly
Nº of square-free chpoly
Nº poly with all real roots
Nº poly with non real roots
Nº Rhapsodic
3 9 6 3 0 0 1
4 27 11 3 0 0 15 81 24 18 0 0 76 243 70 25 0 0 17 729 130 116 0 0 258 2187 341 162 0 0 19 6561 1104 843 0 0 37
10 19683 2380 1637 0 0 711 59049 5934 5880 0 0 121
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 17 / 58
New Challenges
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 18 / 58
New Challenges
Upper Hessenberg Bohemian matrices.We analyze the behaviour of the INFINITY NORM of thecharacteristic polynomials
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 19 / 58
Characteristic polynomials of upper Hessenberg Bohemian matrices
N=4………………………81 matrix
Norms (infinity) of the characteristic polynomials:= {1, 2, 3, 4, 6, 9, 10, 12}…….. 8 different norms
Norm= 1 Norm=2 Norm=3 Norm=4 Norm=6 Norm=9 Norm=10 Norm=12
3 6 18 18 18 6 6 6
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 20 / 58
N=5………………………243 matrices
Set of Norms (infinity):= {1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 19, 22, 25, 26, 28}…….. 15 different norms
Norm= 1 Norm=2 Norm=3 Norm=4 Norm=5 Norm=6 Norm=8 Norm=10 Norm=12
3 6 18 36 18 36 6 42 6
Norm= 13 Norm=19 Norm=22 Norm=25 Norm=26 Norm=28
18 18 18 6 6 6
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 21 / 58
In the good direction…………………..
STEVEN E. THORNTON
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 22 / 58
NORM=10…………….6
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 23 / 58
NORM=10…………….6
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 24 / 58
NORM=10…………….6
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 25 / 58
NORM=10…………….6
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 26 / 58
NORM=10…………….6
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 27 / 58
NORM=10…………….6
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 28 / 58
NORM=1…………….3 NORM=2……………6
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 29 / 58
NORM=3…………….18
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 30 / 58
NORM=4…………….18
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 31 / 58
NORM=6…………….18
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 32 / 58
NORM=9…………….6 NORM=10…………….6
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 33 / 58
NORM=12…………….6
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 34 / 58
N=2N=1 N=3 N=4
N=6N=9 N=10 N=12
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 35 / 58
N=2N=1 N=3 N=4
N=6N=9 N=10 N=12
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 36 / 58
CASE N=5, NORM=1…………….3 NORM=2……..6
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 37 / 58
CASE N=5, NORM=3…………….18
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 38 / 58
CASE N=5, NORM=4…………….36
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 39 / 58
CASE N=5, NORM=5…………….18
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 40 / 58
CASE N=5, NORM=6…………….46
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 41 / 58
CASE N=5, NORM=8…………….6
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 42 / 58
CASE N=5, NORM=10…………….42
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 43 / 58
CASE N=5, NORM=18…………….
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 44 / 58
CASE N=5, NORM=22…………….18
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 45 / 58
CASE N=5, NORM=25…………….6
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 46 / 58
CASE N=5, NORM=26…………….6
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 47 / 58
CASE N=5, NORM=28…………….6
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 48 / 58
Norm Nºmatr
Nº of diff. roots
Nº of charactpoly
Nº of square-free chpoly
Nº poly with allreal roots
Nº poly with n simple real roots
1 3 11 3 2 0 02 6 29 6 6 0 03 18 83 18 16 0 04 36 172 36 34 6 45 18 89 18 18 0 06 36 173 36 34 10 88 6 29 6 6 0 0
10 42 189 42 32 2 013 18 88 18 16 0 019 18 89 18 18 0 022 18 90 18 18 0 025 6 30 6 6 0 026 6 30 6 6 0 028 6 30 6 6 0 0
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 49 / 58
Norm Nºmatr
Nº of diff. roots
Nº of charactpoly
Nº of square-free chpoly
Nº poly with allreal roots
Nº poly with n simple real roots
1 3 11 3 2 0 02 6 29 6 6 0 03 18 83 18 16 0 04 36 172 36 34 6 45 18 89 18 18 0 06 36 173 36 34 10 88 6 29 6 6 0 0
10 42 189 42 32 2 013 18 88 18 16 0 019 18 89 18 18 0 022 18 90 18 18 0 025 6 30 6 6 0 026 6 30 6 6 0 028 6 30 6 6 0 0
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 50 / 58
N=5………………………243 matrices
Set of Norms (infinity):= {1, 2, 3, 4, 5, 6, 8, 10, 13, 19, 3/2, 4/3,5/2,5/3,7/4,10/3,11/3,11/4,11/5,13/2,13/3,13/4,15/5,
13/6,13/7,19/2,19/3,19/4,19/5,19/6,22/5,22/7,22/9,25/11,25/12,28/15}…………………………… 36 different norms
Norm= 1 Norm=2 Norm=3 Norm=4 Norm=5 Norm=6 Norm=8 Norm=10 Norm=123 6 18 36 18 36 6 42 6
Norm= 13 Norm=19 Norm=22 Norm=25 Norm=26 Norm=2818 18 18 6 6 6
Inverses………………….210
Set of Norms (infinity):= {1, 2, 3, 4, 5, 6, 8, 10, 12, 13, 19, 22, 25, 26, 28}…….. 15 different norms
Norm= 1 Norm=2 Norm=3 Norm=4 Norm=5 Norm=6 Norm=8 Norm=10 Norm=132 26 24 20 18 14 2 12 2
Norm= 19 Norm=3/2 Norm=4/3 Norm=5/2 Norm=5/3 Norm=7/4 Norm=10/3 Norm=11/3 Norm=11/42 2 8 10 4 2 6 2 2
Norm= 11/5 Norm=13/2 Norm=13/3 Norm=13/4 Norm=13/5 Norm=13/6 Norm=13/7 Norm=19/2 Norm=19/32 2 4 2 4 4 4 4 2
Norm= 19/4 Norm=19/5 Norm=19/6 Norm=22/5 Norm=22/7 Norm=22/9 Norm=25/11 Norm=25/12 Norm=28/15
4 2 2 2 4 4 2 2 2
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 51 / 58
CASE N=5
N=5………………………243 matrices Inverses………………….210 Rhapsodic………………….42
Set of Norms (Rhapsodic):= {1, 2, 3, 4, 6, 10}…….. 6 different norms
Norm= 1 Norm=2 Norm=3 Norm=4 Norm=6 Norm=102 4 8 14 6 8
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 52 / 58
Matrices 5x5 Inverse Matrices 5x5
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 53 / 58
Rhapsodic (inverses) 5x5Singular Matrices 5x5
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 54 / 58
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 55 / 58
SOME RELATED QUESTIONS
Fixed n and the population:
• How many matrices of each norm are there?
• How many different norms are there?
• Is there any relationship with the norms of the inverse matrices?
• Is there any relationship with the norms of the rhapsodic matrices?
What happen if we change the dimensión n?
What happen if we change the population?
What happen if we change the structure of the matrix?
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 56 / 58
SOME RELATED QUESTIONS
NO ANSWERS YET………………
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 57 / 58
As Queen songs in the "Bohemian Rhapsody"
Is this the real life?Is this just fantasy?No escape from realityOpen your eyes,.............
Open your mind.............
Juana Sendra Pons BOHEMIAN MATRICES: MISE EN PLACE 21 de junio de 2018 58 / 58