BOHEMIAN MATRICES: MISE EN PLACEhigham/conferences/bohemi… · Openyoureyes,........

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

BOHEMIAN MATRICES

Department of Applied MathematicsWestern University Ontario

ROBERT M. CORLESS

Juana Sendra Pons

Rafael Sendra Pons

Eunice Y. S. Chan

Steven Thornton

BOHEMIAN MATRICES

Department of Applied MathematicsWestern University

ROBERT M. CORLESS

Juana Sendra Pons

Rafael Sendra Pons

Eunice Y. S. Chan

Steven Thornton

BOHEMIAN MATRICES

Department of Applied MathematicsWestern University

ROBERT M. CORLESS

Juana Sendra Pons

Rafael Sendra Pons

Eunice Y. S. Chan

Steven Thornton

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.............

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.............

Mise en place

I refer to the Mise en place of Bohemian Matrices, as the task ofarranging the ingredients (experiments and conjectures) before cookthe theory.

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?..............................

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?..............................

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.......

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.......

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

Upper Hessenberg Toeplitz Bohemian matrices

n Nº of matrix

Nº of characteristicpolynomial

Nº sqr-free characteristicpolynomial

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%

New Challenges

Rhapsodic matrices

New Challenges

New Challenges: Toeplitz Bohemian matrices

New Challenges

n Nº of matrix

Nº of charact poly

Nº of square-free chpoly

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

New Challenges

n Nº of matrix

Nº of charact poly

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

New Challenges

Upper Hessenberg Bohemian matrices.We analyze the behaviour of the INFINITY NORM of thecharacteristic polynomials

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

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

In the good direction…………………..

STEVEN E. THORNTON

NORM=10…………….6

NORM=1…………….3 NORM=2……………6

NORM=3…………….18

NORM=4…………….18

NORM=6…………….18

NORM=9…………….6 NORM=10…………….6

NORM=12…………….6

N=2N=1 N=3 N=4

N=6N=9 N=10 N=12

N=2N=1 N=3 N=4

N=6N=9 N=10 N=12

CASE N=5, NORM=1…………….3 NORM=2……..6

CASE N=5, NORM=3…………….18

CASE N=5, NORM=4…………….36

CASE N=5, NORM=5…………….18

CASE N=5, NORM=6…………….46

CASE N=5, NORM=8…………….6

CASE N=5, NORM=10…………….42

CASE N=5, NORM=18…………….

CASE N=5, NORM=22…………….18

CASE N=5, NORM=25…………….6

CASE N=5, NORM=26…………….6

CASE N=5, NORM=28…………….6

Norm Nºmatr

Nº of diff. roots

Nº of charactpoly

Nº poly with allreal 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

Norm Nºmatr

Nº of diff. roots

Nº of charactpoly

Nº poly with allreal 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

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

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

Matrices 5x5 Inverse Matrices 5x5

Rhapsodic (inverses) 5x5Singular Matrices 5x5

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?

SOME RELATED QUESTIONS

NO ANSWERS YET………………

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.............

BOHEMIAN MATRICES: MISE EN PLACEhigham/conferences/bohemi… · Openyoureyes,........

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