Displaying similar documents to “Inverse eigenvalue problem of cell matrices”

Linear preservers of rc-majorization on matrices

Mohammad Soleymani (2024)

Czechoslovak Mathematical Journal

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Let A , B be n × m matrices. The concept of matrix majorization means the j th column of A is majorized by the j th column of B and this is done for all j by a doubly stochastic matrix D . We define rc-majorization that extended matrix majorization to columns and rows of matrices. Also, the linear preservers of rc-majorization will be characterized.

G-matrices, J -orthogonal matrices, and their sign patterns

Frank J. Hall, Miroslav Rozložník (2016)

Czechoslovak Mathematical Journal

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A real matrix A is a G-matrix if A is nonsingular and there exist nonsingular diagonal matrices D 1 and D 2 such that A - T = D 1 A D 2 , where A - T denotes the transpose of the inverse of A . Denote by J = diag ( ± 1 ) a diagonal (signature) matrix, each of whose diagonal entries is + 1 or - 1 . A nonsingular real matrix Q is called J -orthogonal if Q T J Q = J . Many connections are established between these matrices. In particular, a matrix A is a G-matrix if and only if A is diagonally (with positive diagonals) equivalent to a column permutation...

Lower bounds for the largest eigenvalue of the gcd matrix on { 1 , 2 , , n }

Jorma K. Merikoski (2016)

Czechoslovak Mathematical Journal

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Consider the n × n matrix with ( i , j ) ’th entry gcd ( i , j ) . Its largest eigenvalue λ n and sum of entries s n satisfy λ n > s n / n . Because s n cannot be expressed algebraically as a function of n , we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S. Hong, R. Loewy (2004). We also conjecture that λ n > 6 π - 2 n log n for all n . If n is large enough, this follows from F. Balatoni (1969).

On the combinatorial structure of 0 / 1 -matrices representing nonobtuse simplices

Jan Brandts, Abdullah Cihangir (2019)

Applications of Mathematics

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A 0 / 1 -simplex is the convex hull of n + 1 affinely independent vertices of the unit n -cube I n . It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute 0 / 1 -simplices in I n can be represented by 0 / 1 -matrices P of size n × n whose Gramians G = P P have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part D of the transposed inverse P - of P is doubly stochastic and has the...

( 0 , 1 ) -matrices, discrepancy and preservers

LeRoy B. Beasley (2019)

Czechoslovak Mathematical Journal

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Let m and n be positive integers, and let R = ( r 1 , ... , r m ) and S = ( s 1 , ... , s n ) be nonnegative integral vectors. Let A ( R , S ) be the set of all m × n ( 0 , 1 ) -matrices with row sum vector R and column vector S . Let R and S be nonincreasing, and let F ( R ) be the m × n ( 0 , 1 ) -matrix, where for each i , the i th row of F ( R , S ) consists of r i 1’s followed by ( n - r i ) 0’s. Let A A ( R , S ) . The discrepancy of A, disc ( A ) , is the number of positions in which F ( R ) has a 1 and A has a 0. In this paper we investigate linear operators mapping m × n matrices over...

Row Hadamard majorization on 𝐌 m , n

Abbas Askarizadeh, Ali Armandnejad (2021)

Czechoslovak Mathematical Journal

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An m × n matrix R with nonnegative entries is called row stochastic if the sum of entries on every row of R is 1. Let 𝐌 m , n be the set of all m × n real matrices. For A , B 𝐌 m , n , we say that A is row Hadamard majorized by B (denoted by A R H B ) if there exists an m × n row stochastic matrix R such that A = R B , where X Y is the Hadamard product (entrywise product) of matrices X , Y 𝐌 m , n . In this paper, we consider the concept of row Hadamard majorization as a relation on 𝐌 m , n and characterize the structure of all linear operators T : 𝐌 m , n 𝐌 m , n preserving...

Comparison between two types of large sample covariance matrices

Guangming Pan (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Let { X i j } , i , j = , be a double array of independent and identically distributed (i.i.d.) real random variables with E X 11 = μ , E | X 11 - μ | 2 = 1 and E | X 11 | 4 l t ; . Consider sample covariance matrices (with/without empirical centering) 𝒮 = 1 n j = 1 n ( 𝐬 j - 𝐬 ¯ ) ( 𝐬 j - 𝐬 ¯ ) T and 𝐒 = 1 n j = 1 n 𝐬 j 𝐬 j T , where 𝐬 ¯ = 1 n j = 1 n 𝐬 j and 𝐬 j = 𝐓 n 1 / 2 ( X 1 j , ... , X p j ) T with ( 𝐓 n 1 / 2 ) 2 = 𝐓 n , non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of 𝒮 and 𝐒 are different as n with p / n approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the...

Localization of dominant eigenpairs and planted communities by means of Frobenius inner products

Dario Fasino, Francesco Tudisco (2016)

Czechoslovak Mathematical Journal

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We propose a new localization result for the leading eigenvalue and eigenvector of a symmetric matrix A . The result exploits the Frobenius inner product between A and a given rank-one landmark matrix X . Different choices for X may be used, depending on the problem under investigation. In particular, we show that the choice where X is the all-ones matrix allows to estimate the signature of the leading eigenvector of A , generalizing previous results on Perron-Frobenius properties of matrices...

Computing the greatest 𝐗 -eigenvector of a matrix in max-min algebra

Ján Plavka (2016)

Kybernetika

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A vector x is said to be an eigenvector of a square max-min matrix A if A x = x . An eigenvector x of A is called the greatest 𝐗 -eigenvector of A if x 𝐗 = { x ; x ̲ x x ¯ } and y x for each eigenvector y 𝐗 . A max-min matrix A is called strongly 𝐗 -robust if the orbit x , A x , A 2 x , reaches the greatest 𝐗 -eigenvector with any starting vector of 𝐗 . We suggest an O ( n 3 ) algorithm for computing the greatest 𝐗 -eigenvector of A and study the strong 𝐗 -robustness. The necessary and sufficient conditions for strong 𝐗 -robustness are introduced...

On row-sum majorization

Farzaneh Akbarzadeh, Ali Armandnejad (2019)

Czechoslovak Mathematical Journal

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Let 𝕄 n , m be the set of all n × m real or complex matrices. For A , B 𝕄 n , m , we say that A is row-sum majorized by B (written as A rs B ) if R ( A ) R ( B ) , where R ( A ) is the row sum vector of A and is the classical majorization on n . In the present paper, the structure of all linear operators T : 𝕄 n , m 𝕄 n , m preserving or strongly preserving row-sum majorization is characterized. Also we consider the concepts of even and circulant majorization on n and then find the linear preservers of row-sum majorization of these relations on 𝕄 n , m . ...

Controllable and tolerable generalized eigenvectors of interval max-plus matrices

Matej Gazda, Ján Plavka (2021)

Kybernetika

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By max-plus algebra we mean the set of reals equipped with the operations a b = max { a , b } and a b = a + b for a , b . A vector x is said to be a generalized eigenvector of max-plus matrices A , B ( m , n ) if A x = λ B x for some λ . The investigation of properties of generalized eigenvectors is important for the applications. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. In this paper the properties of matrices and vectors with inexact (interval)...

Maps on upper triangular matrices preserving zero products

Roksana Słowik (2017)

Czechoslovak Mathematical Journal

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Consider 𝒯 n ( F ) —the ring of all n × n upper triangular matrices defined over some field F . A map φ is called a zero product preserver on 𝒯 n ( F ) in both directions if for all x , y 𝒯 n ( F ) the condition x y = 0 is satisfied if and only if φ ( x ) φ ( y ) = 0 . In the present paper such maps are investigated. The full description of bijective zero product preservers is given. Namely, on the set of the matrices that are invertible, the map φ may act in any bijective way, whereas for the zero divisors and zero matrix one can write φ as a...

On bilinear forms based on the resolvent of large random matrices

Walid Hachem, Philippe Loubaton, Jamal Najim, Pascal Vallet (2013)

Annales de l'I.H.P. Probabilités et statistiques

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Consider a N × n non-centered matrix 𝛴 n with a separable variance profile: 𝛴 n = D n 1 / 2 X n D ˜ n 1 / 2 n + A n . Matrices D n and D ˜ n are non-negative deterministic diagonal, while matrix A n is deterministic, and X n is a random matrix with complex independent and identically distributed random variables, each with mean zero and variance one. Denote by Q n ( z ) the resolvent associated to 𝛴 n 𝛴 n * , i.e. Q n ( z ) = 𝛴 n 𝛴 n * - z I N - 1 . Given two sequences of deterministic vectors ( u n ) and ( v n ) with bounded Euclidean norms, we study the limiting behavior of the random bilinear form:...