Algebraic multiplicity of the eigenvalues of a tournament matrix

Caen, D. de; Gregory, David A.; Kirkland, Steve; Pullman, Norman J.; Maybee, John S.

doi:10.1016/0024-3795(92)90178-d

Cited by 32 publications

(19 citation statements)

References 9 publications

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“…(Hadamard tournament matrices of order n are coexistent with skew Hadamard matrices of order n + 1, so such H's exist for infinitely many n.) As was shown by de Caen et al [3], the eigenvalues of H are (n -1)/2 (once) and 1/2(-1& iJti)((n -1)/2 times each).…”

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confidence: 88%

“…COROLLARY 1 3. I f A is a generalized tournament matrix of order n with score vector s, and if sts < n2(n -1)/4, then the Perron root of A is larger than ( n -2)/2.Proof If sts < n2(n -1)/4 then d n 2 + 4(n --16sts/n > n -2, so the result…”

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confidence: 90%

“…Let B be the generalized tournament matrix of order n given by and denote its score vector by s. Then s's = n2(n -1)/4 and B is neither irreducible nor primitive. EXAMPLE 2 3. We now exhibit a family of matrices with some negative entries of large modulus which satisfy the hypotheses of Theorem 4.…”

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confidence: 96%

“…The algebraic and geometric multiplicities of an eigenvalue of A with real part -112 are the same (see de Caen et al [3]), so it follows that A has n linearly independent eigenvectors; hence we can express s as a linear combination of them. Since l t u = 0 whenever u is an eigenvector corresponding to an eigenvalue with real part -1/2, the fact that 1 '~u + l t A t u = I'JU -l t u yields s t u = 0 as well.…”

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confidence: 97%

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Hypertournament matrices, score vectors and eigenvalues

Kirkland

1991

Linear and Multilinear Algebra

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Consider a real square matrix A of order n which satisfies A + A' = 3 -I (where J is the all ones matrix) and its score vector, s = A1 (where I is the all ones vector). Here are our main results. If s's < (n3 + 4n(n -1)~)/16, then A has a real positive eigenvalue p with while the other eigenvalues satisfy A has eigenvalues p and X such that and if and only if A has n -2 eigenvalues with real part -112. If sfs 5 n2(n -1)/4 then for any eigenvalue A, ( I~( x ) )~ < (n2 + 2n -4)/8 -t r a c e (~~) / 2 . Further, if s's < n2(n -1)/4 and trace(^^) 2 -(n2 -10n +10)/4 then a Perron-Frobenius result holds for A. A consequence of this is that if such an A is nonnegative as well, and if n 2 9, then A is irreducible and primitive.

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confidence: 88%

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confidence: 90%

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confidence: 96%

mentioning

confidence: 97%

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Hypertournament matrices, score vectors and eigenvalues

Kirkland

1991

Linear and Multilinear Algebra

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“…In [3] it is shown that if A is a doubly regular tournament matrix, then its eigenvalues consist of n−1 2 (of algebraic multiplicity one, and having 1 as a corresponding right eigenvector) and…”

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confidence: 99%

Girth and subdominant eigenvalues for stochastic matrices

Kirkland

2005

ELA

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Abstract. The set S(g, n) of all stochastic matrices of order n whose directed graph has girth g is considered. For any g and n, a lower bound is provided on the modulus of a subdominant eigenvalue of such a matrix in terms of g and n, and for the cases g = 1, 2, 3 the minimum possible modulus of a subdominant eigenvalue for a matrix in S(g, n) is computed. A class of examples for the case g = 4 is investigated, and it is shown that if g > 2n/3 and n ≥ 27, then for every matrix in S(g, n), the modulus of the subdominant eigenvalue is at least ( 1 5 ) 1/(2 n/3 ) .

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