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Overview . Eigenspaces of Tournament Matrices 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Definitions and Examples Overview Motivational Questions Overview I. Preliminaries II. Basic Tournament Properties III. The Brualdi-Li Perron Eigenspace IV. Purely Imaginary Eigenvalues V. Open Questions 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Definitions and Examples Overview Motivational Questions Definitions A ∈ M n ( R ) is a tournament matrix if is it satisfies A + A t = J n − I n , A ≥ 0 , A ◦ A = A . A ∈ M n ( R ) is a generalized tournament matrix if is it satisfies A + A t = J n − I n , A ≥ 0 . A ∈ M n ( R ) is a 1-hypertournament matrix if is it satisfies A + A t = J n − I n . A ∈ M n ( R ) is a h-hypertournament matrix if is it satisfies A + A t = hh t − I n . 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Definitions and Examples Overview Motivational Questions Examples Example Consider the following matrices: − 1 1       4 10 6 0 0 0 0 3 2 2 A = − 1 4 5 B = 0 0 2 C = 0 1  .      3 3 4 4 0 − 1 0 0 1 0 A is a 3 -hypertournament matrix, where 3 = 3 · 1 3 . B is a h -hypertournament matrix, where h = [ 0 , 1 , 1 ] t . (Note: ± i √ 2 ∈ σ ( B ) ) C is a generalized tournament matrix. 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Definitions and Examples Overview 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? 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Matrix Notations General Information Spectrum Notations Operations Notations Notations M n n × n complex matrices. M n ( R ) n × n real matrices. I n the n × n identity matrix. the n × n all-ones matrix. J n O k , m the n × m zero matrix. the n × 1 all-ones vector. 1 n the k-th standard basis vector for R n e k 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Matrix Notations General Information Spectrum Notations Operations Notations Notations σ ( A ) multi-set of eigenvalues of A ∈ M n p A ( t ) characteristic polynomial of A ∈ M n tr ( A ) trace of A ∈ M n det ( A ) determinant of A ∈ M n ρ ( A ) = max λ ∈ σ ( A ) | λ | spectral-radius of A ∈ M n 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Matrix Notations General Information Spectrum Notations Operations Notations Notations A t transpose of A ∈ M n A ∗ conjugate transpose of A ∈ M n A − 1 inverse of A ∈ M n √ Euclidean norm for x ∈ C n � x � 2 = x ∗ x 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Nonnegative Matrices Basics Nonnegative Matrix Definition A matrix A = [ a ij ] ∈ M n is said to be nonnegative if a ij ≥ 0 for i , j = 1 , . . . , n , and it is denoted as A ≥ 0. 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 ∈ M n ( R ) , A ≥ 0 , corresponding to ρ ( A ) which is simple. When x and y are normalized so that 1 t n x = 1 t n y = 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 . 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Nonnegative Matrices Basics Perron-Frobienus Theorem (Perron-Frobienus) Let A ∈ T n be irreducible and nonnegative, then i) ρ := ρ ( A ) > 0 ii) ρ ∈ σ ( A ) iii) ∃ x > 0 s.t. Ax = ρ x iv) ρ is simple 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Nonnegative Matrices Basics Bounding ρ ( A ) Theorem Let A = [ a jk ] ∈ M n ( R ) be nonnegative. Then, for any positive vector x ∈ R n we have n n 1 1 � � min a kj x j ≤ ρ ( A ) ≤ max a kj x k x k 1 ≤ k ≤ n 1 ≤ k ≤ n j = 1 j = 1 and n n 1 a kj a kj � � ≤ ρ ( A ) ≤ max 1 ≤ j ≤ n x j min 1 ≤ j ≤ n x j . x k x k x k k = 1 k = 1 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Definition, Basic Properties Tournaments Ranking Schemes Regular and Almost Regular Perron Value: Maximizing Condition Tournament Matrices: Example Example   � 0 1 0 � 0 0 0 1 1 0 0 0   A 1 = 0 0 1 A 2 = 1 1 0 0 1 0 0 0 1 1 0 1 1 2 2 3 3 4 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Definition, Basic Properties Tournaments Ranking Schemes Regular and Almost Regular 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 a ij = 1; furthermore, because j losses under player i then a ji = 0. The i th column sum is the number of losses for player i ; whereas, the i th row sum is the number of wins for player i . 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Definition, Basic Properties Tournaments Ranking Schemes Regular and Almost Regular Perron Value: Maximizing Condition Score Vector Definition A vector s ∈ R is said to be the score vector for a given tournament matrix T ∈ T n provided that s = T 1 , where 1 is the n × 1 all-ones vector. 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Definition, Basic Properties Tournaments Ranking Schemes Regular and Almost Regular Perron Value: Maximizing Condition Score Vector Example     0 0 0 1 1 1 0 0 0 1   1 =   T 1 = 1 1 0 0 2 0 1 1 0 2     0 0 0 1 2 1 0 0 0 2 1 t T = 1 t   =   1 1 0 0 1 0 1 1 0 1 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Definition, Basic Properties Tournaments Ranking Schemes Regular and Almost Regular Perron Value: Maximizing Condition Who is the best player? How can we rank the players? 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Definition, Basic Properties Tournaments Ranking Schemes Regular and Almost Regular Perron Value: Maximizing Condition Ranking Schemes Score Ranking Players are ranked according to the score vector. 1 Strengths of a player are represented by number of wins in the 2 score vector. Kendall-Wei Ranking Players are ranked according to the (right) Perron vector. 1 Strengths of a player are represented by the (right) Perron vector. 2 Ramanajucharyula Ranking Players are ranked according the right and left Perron vectors. 1 Strengths and Weaknesses are represented by the right and left 2 Perron vectors, respectively. Next 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Definition, Basic Properties Tournaments Ranking Schemes Regular and Almost Regular 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 = T 1 . Notice: the i th entry in Ts represents the sum of the strengths of the players that player i defeats. r 2 = Ts = T ( T 1 ) = T 2 1 . Repeat this process for T 2 , T 3 , . . . in order to obtain the following sequence r 3 = Tr 2 = T 2 s , . . . . r 1 = s , r 2 = Ts , � � ∞ r k = T k − 1 s k = 1 Power Method! Return 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Definition, Basic Properties Tournaments Ranking Schemes Regular and Almost Regular 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: x k , y k where Tx = ρ x , y t T = ρ y t , and ρ := ρ ( A ) for a given tournament matrix T . Return 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Definition, Basic Properties Tournaments Ranking Schemes Regular and Almost Regular Perron Value: Maximizing Condition Perron Value Relation Theorem Let T and � T be two n × n tournament matrices with Perron vectors x, y be left Perron vectors of T, � x, respectively. Let also y, � � T, respectively. Then the following are equivalent. (a) ρ ( T ) ≤ ρ ( � ( b ) � x � 2 ≥ � � ( c ) � y � 2 ≥ � � T ) x � 2 y � 2 Furthermore, either in all of the above statements the inequalities are strict, or they all hold as equalities. Skip Proof 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices

Definition, Basic Properties Tournaments Ranking Schemes Regular and Almost Regular Perron Value: Maximizing Condition Proof Let T ∈ T n be a tournament matrix, and x , y ∈ R n be right and left Perron vectors (respectfully). T + T t = J n − I n x t � T + T t � x = x t ( J n − I n ) x = x t 11 t x − x t x ( 2 ρ ) x t x = 1 − x t x 1 − 1 ρ = 2 � x � 2 2 2 Notice that ρ and � x � 2 2 are indirectly proportional ; therefore, ρ increases ⇔ � x � 2 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) . 0 1 2 3 4 5 6 7 8 James Burk Eigenspaces of Tournament Matrices