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Overview

.

Eigenspaces of Tournament Matrices

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Overview Definitions and Examples Motivational Questions

Overview

• I. Preliminaries
• II. Basic Tournament Properties
• III. The Brualdi-Li Perron Eigenspace
• IV. Purely Imaginary Eigenvalues
• V. Open Questions

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Overview Definitions and Examples Motivational Questions

Definitions

A ∈ Mn (R) is a tournament matrix if is it satisfies A + At = Jn − In, A ≥ 0, A ◦ A = A. A ∈ Mn (R) is a generalized tournament matrix if is it satisfies A + At = Jn − In, A ≥ 0. A ∈ Mn (R) is a 1-hypertournament matrix if is it satisfies A + At = Jn − In. A ∈ Mn (R) is a h-hypertournament matrix if is it satisfies A + At = hht − In.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Overview Definitions and Examples Motivational Questions

Examples

Example Consider the following matrices: A =   4 10 6 −1 4 5 3 4 4   B =   − 1

2

2 −1   C =  

1 3 2 3

1 1   . A is a 3-hypertournament matrix, where 3 = 3 · 13. B is a h-hypertournament matrix, where h = [0, 1, 1]t. (Note: ±i

√ 2 ∈ σ(B))

C is a generalized tournament matrix.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Overview Definitions and Examples Motivational Questions

Questions

How do you rank players in a given tournament? When do you obtain the maximal spectral radius? Do tournament matrices have nonzero purely imaginary eigenvalues?

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

General Information Matrix Notations Spectrum Notations Operations Notations

Notations

Mn n × n complex matrices. Mn (R) n × n real matrices. In the n × n identity matrix. Jn the n × n all-ones matrix. Ok,m the n × m zero matrix. 1n the n × 1 all-ones vector. ek the k-th standard basis vector for Rn

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

General Information Matrix Notations Spectrum Notations Operations Notations

Notations

σ(A) multi-set of eigenvalues of A ∈ Mn pA(t) characteristic polynomial of A ∈ Mn tr (A) trace of A ∈ Mn det (A) determinant of A ∈ Mn ρ(A) = max

λ∈σ(A) |λ|

spectral-radius of A ∈ Mn

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

General Information Matrix Notations Spectrum Notations Operations Notations

Notations

At transpose of A ∈ Mn A∗ conjugate transpose of A ∈ Mn A−1 inverse of A ∈ Mn x2 = √ x∗x Euclidean norm for x ∈ Cn

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Nonnegative Matrices Basics

Nonnegative Matrix

Definition A matrix A = [aij] ∈ Mn is said to be nonnegative if aij ≥ 0 for i, j = 1, . . . , n, and it is denoted as A ≥ 0.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Nonnegative Matrices Basics

Perron Vectors

Definition Let x and y denote the right and left eigenvectors, respectively, of A ∈ Mn (R) , A ≥ 0, corresponding to ρ(A) which is simple. When x and y are normalized so that 1t

nx = 1t ny = 1

then x and y are unique and they are respectively referred to as the (right) Perron vector and the left Perron vector of A.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Nonnegative Matrices Basics

Perron-Frobienus

Theorem (Perron-Frobienus) Let A ∈ Tn be irreducible and nonnegative, then i) ρ := ρ(A) > 0 ii) ρ ∈ σ(A) iii) ∃ x > 0 s.t. Ax = ρx iv) ρ is simple

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Nonnegative Matrices Basics

Bounding ρ(A)

Theorem Let A = [ajk] ∈ Mn (R) be nonnegative. Then, for any positive vector x ∈ Rn we have min

1≤k≤n

1 xk

n

• j=1

akjxj ≤ ρ(A) ≤ max

1≤k≤n

1 xk

n

• j=1

akj and min

1≤j≤n xj

1 xk

n

• k=1

akj xk ≤ ρ(A) ≤ max

1≤j≤n xj n

• k=1

akj xk .

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Tournaments Regular and Almost Regular Definition, Basic Properties Ranking Schemes Perron Value: Maximizing Condition

Tournament Matrices: Example

Example A1 = 0 1 0 0 0 1 1 0 0

• A2 =

  0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0   1 2 3 1 2 3 4

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Tournaments Regular and Almost Regular Definition, Basic Properties Ranking Schemes Perron Value: Maximizing Condition

Round-Robin Tournament

In a round-robin tournament n-players play against the other (n − 1)-players. If player i wins over player j then aij = 1; furthermore, because j losses under player i then aji = 0. The ith column sum is the number of losses for player i; whereas, the ith row sum is the number of wins for player i.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Tournaments Regular and Almost Regular Definition, Basic Properties Ranking Schemes Perron Value: Maximizing Condition

Score Vector

Definition A vector s ∈ R is said to be the score vector for a given tournament matrix T ∈ Tn provided that s = T1, where 1 is the n × 1 all-ones vector.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Tournaments Regular and Almost Regular Definition, Basic Properties Ranking Schemes Perron Value: Maximizing Condition

Score Vector

Example T1 =   0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0   1 =   1 1 2 2   1tT = 1t   0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0   =   2 2 1 1  

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Tournaments Regular and Almost Regular Definition, Basic Properties Ranking Schemes Perron Value: Maximizing Condition

Who is the best player?

How can we rank the players?

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Tournaments Regular and Almost Regular Definition, Basic Properties Ranking Schemes Perron Value: Maximizing Condition

Ranking Schemes

Score Ranking

1

Players are ranked according to the score vector.

2

Strengths of a player are represented by number of wins in the score vector. Kendall-Wei Ranking

1

Players are ranked according to the (right) Perron vector.

2

Strengths of a player are represented by the (right) Perron vector. Ramanajucharyula Ranking

1

Players are ranked according the right and left Perron vectors.

2

Strengths and Weaknesses are represented by the right and left Perron vectors, respectively.

Next 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Tournaments Regular and Almost Regular Definition, Basic Properties Ranking Schemes Perron Value: Maximizing Condition

Score vector leading to the Kendall-Wei scheme

Let T be a given tournament matrix, and consider its score vector s = T1. Notice: the ith entry in Ts represents the sum of the strengths of the players that player i defeats. r2 = Ts = T(T1) = T 21. Repeat this process for T 2, T 3, . . . in order to obtain the following sequence r1 = s, r2 = Ts, r3 = Tr2 = T 2s, . . . .

• rk = T k−1s

∞

k=1

Power Method!

Return 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Tournaments Regular and Almost Regular Definition, Basic Properties Ranking Schemes Perron Value: Maximizing Condition

Ramanajucharyula ranking

Same idea as behind the Kendall-Wei scheme, but it now incorporates the weakness of the players. Examines the ratio, strength-to-weakness: xk yk , where Tx = ρx, ytT = ρyt, and ρ := ρ(A) for a given tournament matrix T.

Return 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Tournaments Regular and Almost Regular Definition, Basic Properties Ranking Schemes Perron Value: Maximizing Condition

Perron Value Relation

Theorem Let T and T be two n × n tournament matrices with Perron vectors x,

• x, respectively. Let also y,

y be left Perron vectors of T, T,

• respectively. Then the following are equivalent.

(a) ρ(T) ≤ ρ( T) (b) x2 ≥ x2 (c) y2 ≥ y2 Furthermore, either in all of the above statements the inequalities are strict, or they all hold as equalities.

Skip Proof 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Tournaments Regular and Almost Regular Definition, Basic Properties Ranking Schemes Perron Value: Maximizing Condition

Proof

Let T ∈ Tn be a tournament matrix, and x, y ∈ Rn be right and left Perron vectors (respectfully). T + T t = Jn − In xt T + T t x = xt (Jn − In) x = xt11tx − xtx (2ρ)xtx = 1 − xtx ρ = 1 2x2

2

− 1 2 Notice that ρ and x2

2 are indirectly proportional; therefore, ρ

increases ⇔ x2

2 decreases, thereby showing the equivalence of (a)

and (b) In parallel, same analysis can be used for y instead of x, thereby, showing the equivalence of (a) and (c).

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Tournaments Regular and Almost Regular Definition Brualdi-Li Matrix

Regular and Almost Regular

Definition Let T ∈ Tn be a given tournament matrix. If n is odd and all of the row sums of T are equal (n − 1)/2, then T is called regular. If n is even, and half of the row sums of T equal n/2 and the rest equal (n − 2)/2, then T is called almost regular.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Tournaments Regular and Almost Regular Definition Brualdi-Li Matrix

Commonalities

Theorem Let A ∈ T2m+1 be regular and let B ∈ T2(m+1) be almost regular, where m > 1. Then A and B share the following properties.

1

A and B are nonsingular

2

A and B are irreducible

3

A and B are primitive

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Tournaments Regular and Almost Regular Definition Brualdi-Li Matrix

Almost Regular: First Kind

Definition Given T ∈ Tm, and construct the tournament matrix A ∈ T2m as A =

• T

T t T t + Im T

• .

The matrix A will be referred to as an almost regular of the first kind, which is generated by T.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Tournaments Regular and Almost Regular Definition Brualdi-Li Matrix

The Brualdi-Li Matrix

Definition The Brualdi-Li tournament matrix is defined as B2m =

• L

Lt Lt + Im L

• ,

where L denotes the m × m strictly lower triangular tournament matrix, i.e. all of the entries below the main diagonal are equal to one.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Tournaments Regular and Almost Regular Definition Brualdi-Li Matrix

Brualdi-Li Examples

Example B4 =   0 0 0 1 1 0 0 0 1 1 0 0 0 1 1 0   B6 =      0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 0      1.3660 < ρ (B4) = 1.3953 < 1.4063 2.4142 < ρ (B6) = 2.4340 < 2.4375

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Tournaments Regular and Almost Regular Definition Brualdi-Li Matrix

Regular: Perron Vectors

Recall that regular tournaments are normal. Theorem Let A ∈ Mn be a normal matrix. Then x ∈ Cn is a right eigenvector of A corresponding to the eigenvalue of λ of A if and only if x is a left eigenvector of A corresponding to λ.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Tournaments Regular and Almost Regular Definition Brualdi-Li Matrix

Almost Regular: Perron Vectors

Is there an analog for the Perron vectors for a given almost regular?

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Tournaments Regular and Almost Regular Definition Brualdi-Li Matrix

Brualdi-Li: Perron Vectors

Theorem Let x and y respectively be the right and left Perron vector for B2m. Then y = C2mx. Why? Because Ct

2mBt 2mC2m = B2m, where

C2m = Rm Om Om Rm

• and Rm is the m × m reverse identity matrix.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

The Structure of the Brualdi-Li

What makes the Brualdi-Li interesting?

1

Toeplitz

2

Persymmetic

3

Almost Circulant

4

We have explicitly calculated the inverse.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

Brualdi-Li Inverse

Let Ik and Jk denote the k × k identity matrix and all-ones matrix,

• respectively. Let also m ≥ 2 and consider the m × m anti-diagonal

matrix P = [pij] defined by pij = 1 if i + j = m + 1

• therwise,

Define the matrices A = Jm−1 − (m − 1)Im−1 1m−1 0t

m−1

1 − m

• C =

0m−1 Jm−1 − (m − 1)Im−1 m − 1 0t

m−1

• .

Then, B−1

2m =

1 m − 1

• A

C Jm − (m − 1)Im PAtP

• .

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

Proof Outline

A matrix is given, and then it is shown that it works! How was this pattern derived? Seeing a pattern after computing lots of examples.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

The Brualdi-Li Conjecture

Conjecture If T ∈ M2m is a tournament matrix, then ρ(T) ≤ ρ(B2m).

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

The Brualdi-Li Conjecture Affirmation

Theorem If T ∈ M2m is a tournament matrix, then ρ(T) ≤ ρ(B2m).

• S. W. Drury. Solution of the Conjecture of Brualdi and Li. Linear

Algebra Appl., 436: 3392 3399, 2012.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

Eigenvalues for First-Kind Almost Regular

Theorem Let v = [vj] ∈ Rm and w = [wj] ∈ Rm so that x = v w

• ∈ R2m is an

eigenvector of the tournament matrix T =

• T

T t T t + Im T

• corresponding to λ. Then,

λ = 1t

2mx − wk

vk + wk .

Skip Proof 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

Proof

First notice that T x = λx yields λv λw

• =
• T

T t T t + Im T v w

• =
• Tv + T tw

T tv + v + Tw

• =

(Jm − Im − T t)v + T tw T tv + v + Tw

• .

Equating these blocks in order to obtain the following results.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

Proof

Jmv + T tw + Tw = λ(w + v) = ⇒ Jmv + (Jm − Im)w = λ(w + v) = ⇒ 1

• 1t(v + w)
• − w = λ(w + v)

= ⇒ 1t1

• 1t(w + v)
• − 1tw = λ
• 1t(w + v)
• =

⇒ m − 1tw 1t(w + v) = λ.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

Proof

Furthermore, the equation 1

• 1t(v + w)
• − w = λ(w + v)

yields λ(wk + vk) = 1

• 1t(v + w)
• − wk

= ⇒ λ = 1

• 1t(v + w)
• − wk

(wk + vk) . And since 1t

2m = 1t m(v + w) then we are done.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

Perron Eigenvectors

Notation Let v = [vj] ∈ Rm and w = [wj] ∈ Rm so that x = v w

• ∈ R2m is the

Perron vector of B2m. Furthermore, y ∈ R2m shall denote the left-Perron vector.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

Eigenvector: Ordering

Theorem vm < vm−1 < . . . < v1 < w1 < w2 < . . . < wm. Kendall-Wei Interpretation: The player corresponding to wm, i.e. player 2m is the strongest, followed (in order) by players 2m − 1, 2m − 2, . . . , m + 1, 1, 2, . . . , m.

Skip Proof 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

Proof Outline

Since B2mx = ρx, we have that for k = 1, 2, . . . , m − 1, ρvk =

k−1

• j=1

vj +

m

• j=k+1

wj from which it follows that for all k = 1, 2, . . . , m − 1, ρvk − ρvk+1 =  

k−1

• j=1

vj +

m

• j=k+1

wj   −  

k

• j=1

vj +

m

• j=k+2

wj   = wk+1 − vk. Applying a proposition in the paper, yields for k = 1, 2, . . . , m, vk+1 < vk.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

Proof Outline

Because wk = 1 − ρvk ρ + 1 for each k = 1, . . . , m and since 0 ≤ vk+1 < vk for k = 1, 2, . . . , m − 1, then wk < wk+1 for k = 1, 2, . . . , m − 1. Then we apply a result from Kirkland that states vj < wk for j, k = 1, . . . , m in order to complete the proof.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

Eigenvector: Explicit

Theorem v1 = (m − 1)(ρ + 1) − ρ2 ρ2 w1 = ρ + 1 − m ρ vm = ρ + 1 − m ρ + 1 wm = 1 + mρ − ρ2 (ρ + 1)2 vk+1 = vk(ρ + 1)2 − 1 ρ2 wk+1 = wk(ρ + 1)2 − 1 ρ2 for k = 1, 2, . . . , m − 1. Recall: Bx = ρx and ytB = ρyt.

Skip Proof 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

Proof Outline

As a proof, we make use of the fact that ρvk =

k−1

• j=1

vj +

m

• j=k+1

wj. Then we examine the ratios wℓ − vℓ wj − vj = ρ + 1 ρ 2(ℓ−j) . Then we apply use a fact from Kirkland [?] that states 2ρ2 − 2ρ(m − 1) − (m − 1) = ρ + 1 ρ 2m + 1 −1 . And, then lots of algebra manipulations.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

Interlaced Relationship: Strength to Weakness

Theorem xm ym < x1 y1 < xm−1 ym−1 < x2 y2 < · · · < x⌈m/2⌉ y⌈m/2⌉ < 1 < < x2m−⌈m/2⌉ + 1 y2m−⌈m/2⌉ + 1 < · · · < xm+2 ym+2 < x2m y2m < xm+1 ym+1 .

Skip Proof 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

Proof Outline

1

Define the function f(k) = yk/xk, for k = 1, . . . , m.

2

Show f(m) > f(m − j), for j = 1, . . . , m − 1.

3

Show f(j) > f(m − j), for j = 1, . . . , ⌈m/2⌉.

4

Show f(m − j) > f(j + 1), for j = 1, . . . , ⌈m/2⌉.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

Interlaced Relationship

Example After computing the right Perron, x, and the left Perron, y, vectors for B12 we have the following interlaced relationship x6 y6 = 0.8454 < x1 y1 = 0.8738 < x5 y5 = 0.8761 < x2 y2 = 0.8910 < < x4 y4 = 0.8927 < x3 y3 = 0.8973 < 1 < x10 y10 = 1.1144 < x9 y9 = 1.1202 < x11 y11 = 1.1224 < x8 y8 = 1.1414 < < x12 y12 = 1.1444 < x7 y7 = 1.1829.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research First-Kind Almost Regular Perron Eigenspace of B2m Difference in Ranking Schemes

Strength of Players: Increasing to Decreasing

Example Score Vector Players 1 to 6 are equal strength, and Players 7 to 12 are equal

• strength. Players 7 to 12 are stronger than players 1 to 6

Kendall-Wei Ranking 12, 11, 10, 9, 8, 7, 1, 2, 3, 4, 5, 6. Ramanajucharyula Ranking 7, 12, 8, 11, 9, 10, 3, 4, 2, 5, 1, 6.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research Differential Equation Power Iterations Perturbations of B2m + γJ2m

ytx

Theorem Let v = [vj] ∈ Rm and w = [wj] ∈ Rm (m ≥ 2) so that x = v w

• ∈ R2m is the Perron vector of B2m and let ρ = ρ(B2m);

furthermore, let y denote the left Perron vector of B2m. Then ytx = 2

• m +

1 2ρ + 1 − m2(m + 1) ρ + 1 + m2(m − 1) ρ

• .

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research Differential Equation Power Iterations Perturbations of B2m + γJ2m

Separable Differential Equation

∂ρ ∂bij = yi xj ytx . By choosing i = m and j = m + 1, leads to ⇒ ∂ρ ∂bij = vmw1 ytx . Replacing the (m, m + 1) entry in B2m by the variable t, then when t = 0 we have

dρ dt

• t=0

= (ρ − m + 1)2 ρ(ρ + 1)

• 2(m +

1 2ρ + 1 − m2(m + 1) ρ + 1 + m2(m − 1) ρ ) −1

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research Differential Equation Power Iterations Perturbations of B2m + γJ2m

Separable Differential Equation

Solving this separable differential equation yields:

2mρ − ln(2ρ + 1) (2m − 1)2 − 2(m − 1)

• 4m2 − 2m + 1
• ln(ρ − m + 1)

(2m − 1)2 − 2m (m − 1) (2m − 1) (ρ − m + 1) = C

Can we independently evaluate C?

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research Differential Equation Power Iterations Perturbations of B2m + γJ2m

Power Method

Choose a vector z ∈ R2m with a component in the direction of the Perron vector x for B2m. lim

k→∞

Bkz 1t

2mBkx

= x lim

k→∞

Bk+1z 1t

2mBkx

= Bx = ρx lim

k→∞

em+1Bk+1z 1t

2mBkx

= ρem+1x = ρw1 = ρ + 1 − m

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research Differential Equation Power Iterations Perturbations of B2m + γJ2m

Power Method

ρ = (m − 1) + lim

k→∞

em+1Bk+1z 1t

2mBkx

What is a good choice for z that will allow us to “easily” compute this limit?

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research Differential Equation Power Iterations Perturbations of B2m + γJ2m

Eigenvector: Explicit

Theorem v1 =

• (m − 1)(ρ + 1) − ρ2 + γ
• 2m(ρ + 1) + (ρ − 1)
• ρ2
• 1t

m(v + w)

• vm =

ρ + 1 − m − γ(2m − 1) ρ + 1 1t

m(v + w)

• vk+1 = (ρ + 1)2vk − (2γ + 1)
• 1t

m(v + w)

• ρ2

for k = 1, 2, . . . , m − 1.

Recall: Bx = ρx and ytB = ρyt where ρ := ρ(B) and B = B2m + γJ2m.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Perron Eigenspace of the Brualdi-Li Future Research Differential Equation Power Iterations Perturbations of B2m + γJ2m

Eigenvector: Explicit

Theorem w1 = ρ + 1 − m − γ(2m − 1) ρ 1t

m(v + w)

• wm =

mρ + 1 − ρ2 + γ(ρ + 2mρ + 2) (ρ + 1)2 1t

m(v + w)

• wk+1 = wk(ρ + 1)2 − (1 + 2γ)
• 1t

m(v + w)

• ρ2

for k = 1, 2, . . . , m − 1.

Recall: Bx = ρx and ytB = ρyt where ρ := ρ(B) and B = B2m + γJ2m.

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Perron Eigenspace of the Brualdi-Li Future Research Differential Equation Power Iterations Perturbations of B2m + γJ2m

Closed Form: ρ (B2m + γJ2m)

Theorem Let C = B2m + γJ2m where γ ∈ C, and (λ, x) be an eigenpair of C; furthermore, define v, w ∈ Cm such that x = v w

• . Then,

ρ + 1 ρ 2m + 1 −1 = (2ρ + 1)2 2(2γ + 1) − m(2ρ + 1) + 1/2, where ρ := ρ (B2m + γJ2m).

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Open Questions Brualdi-Li Tally ρ

• B2m
• and ρ
• B2m + γJ2m
• Definition

Definition If T = T (A, B) is a tally tournament where B = Bm and A ∈ Tn is a regular tournament, then we call T the Brualdi-Li tally (tournament) of order m.

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Open Questions Brualdi-Li Tally ρ

• B2m
• and ρ
• B2m + γJ2m
• Brualdi-Li Tally Conjecture

Conjecture Let nm ∈ N be a nontrivial composite even integer and η be the smallest prime factor of nm. Let T ∈ Tnm be any tournament, which is not permutationally similar to Bnm. If there exists a Brualdi-Li tally, T , of order (nm)/η then ρ(T) ≤ ρ (T ) < ρ (Bnm) .

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Open Questions Brualdi-Li Tally ρ

• B2m
• and ρ
• B2m + γJ2m
• .

What is the connection between ρ (B2m) and ρ (B2m + γJ2m)? Recall ρ + 1 ρ 2m + 1 −1 = (2ρ + 1)2 2(2γ + 1) − m(2ρ + 1) + 1/2, where ρ := ρ (B2m + γJ2m).

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Traceless Definition and Notations Results

Definition

Definition A matrix A ∈ Mn is said to be traceless if tr (A) = 0.

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Traceless Definition and Notations Results

Traceless Notation

Notation Let A ∈ Mn be a traceless matrix such that ρ(A) ∈ σ(A). The real and imaginary parts of the eigenvalues of A are denoted by αk = Re (λk) , and βk = Im (λk) , where λk ∈ σ(A). We will order the real parts as α1 ≤ α2 ≤ · · · ≤ αn.

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Traceless Definition and Notations Results

Traceless Notation

Notation The variables π, τ, ℓ will be reserved to denote the following quantities. π denotes the number of eigenvalues such that Re (λ) = ρ(A). τ denotes the number of eigenvalues such that 0 < Re (λ) = ρ(A). ℓ denotes the number of eigenvalues such that Re (λ) = 0.

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Traceless Definition and Notations Results

Traceless Notation

Notation α1 ≤ · · · ≤ αn−τ−ℓ−π

• n−τ−ℓ−π

< αn−τ−ℓ−π+1 = · · · = αn−τ−π = 0

• ℓ

< αn−τ−π+1 ≤ · · · ≤ αn−π

• τ

< αn−π+1 = · · · = αn

• π

. In addition, we will also define αM = max

1≤k≤n−τ−ℓ−π |αk|.

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Traceless Definition and Notations Results

Conditions

Theorem Let A ∈ Mn (R) be a real traceless matrix. If αn ∈ σ(A), 2 < n, and κ(n − 2)αM ≤ αn, where κ ∈ R. If 1 ≤ πκ then the spectrum of A has the following properties: a) The spectrum of A does not contain a nonzero purely imaginary eigenvalue. b) If λ ∈ σ(A) then either Re (λ) < 0 or Re (λ) = αn. c) Either π = 1 or π = 2. Furthermore, if 0 ∈ σ(A) then amA(0) = 1.

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Traceless Definition and Notations Results

Necessary Condition

Theorem If there exists a tournament matrix, A ∈ Tn, such that it contains a nonzero purely imaginary eigenvalue, then we can infer the following about A. (a) A must be an irreducible matrix such that ρ(A) < n − 2 2 . (b) A cannot contain a regular submatrix of size (n − 1).

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Traceless Definition and Notations Results

.

Thank You.

1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices