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We consider an accessibility index for the states of a discrete-time, ergodic, homogeneous Markov chain on a finite state space; this index is naturally associated with the random walk centrality introduced by Noh and Reiger (2004) for a random walk on a connected graph. We observe that the vector of accessibility indices provides a partition of Kemeny's constant for the Markov chain. We provide three characterizations of this accessibility index: one in terms of the first return time to the state...

Range symmetric matrices in Minkowski space.

Meenakshi, A.R. (2000)

Bulletin of the Malaysian Mathematical Sciences Society. Second Series

Ranges of Sylvester maps and a minimal rank problem.

Ran, Andre C.M., Rodman, Leiba X. (2010)

ELA. The Electronic Journal of Linear Algebra [electronic only]

Rank and inertia of submatrices of the Moore-Penrose inverse of a Hermitian matrix.

Tian, Yongge (2010)

ELA. The Electronic Journal of Linear Algebra [electronic only]

Rank and perimeter preserver of rank-1 matrices over max algebra

Seok-Zun Song, Kyung-Tae Kang (2003)

Discussiones Mathematicae - General Algebra and Applications

For a rank-1 matrix $A = a \otimes b^{t}$ over max algebra, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over max algebra. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices if and only if it has the form T(A) = U ⊗ A ⊗ V, or $T (A) = U \otimes A^{t} \otimes V$ with some monomial matrices U and V.

Rank decomposition in zero pattern matrix algebras

Harm Bart, Torsten Ehrhardt, Bernd Silbermann (2016)

Czechoslovak Mathematical Journal

For a block upper triangular matrix, a necessary and sufficient condition has been given to let it be the sum of block upper rectangular matrices satisfying certain rank constraints; see H. Bart, A. P. M. Wagelmans (2000). The proof involves elements from integer programming and employs Farkas' lemma. The algebra of block upper triangular matrices can be viewed as a matrix algebra determined by a pattern of zeros. The present note is concerned with the question whether the decomposition result referred...

Rank of matrices and the Pexider equation.

Kalinowski, Józef (2006)

Beiträge zur Algebra und Geometrie

Rank of tensors of $ℓ$ -out-of- $k$ functions: An application in probabilistic inference

Jiří Vomlel (2011)

Kybernetika

Bayesian networks are a popular model for reasoning under uncertainty. We study the problem of efficient probabilistic inference with these models when some of the conditional probability tables represent deterministic or noisy $ℓ$ -out-of- $k$ functions. These tables appear naturally in real-world applications when we observe a state of a variable that depends on its parents via an addition or noisy addition relation. We provide a lower bound of the rank and an upper bound for the symmetric border rank...

Rank, trace and determinant in Banach algebras: generalized Frobenius and Sylvester theorems

Gareth Braatvedt, Rudolf Brits, Francois Schulz (2015)

Studia Mathematica

As a follow-up to a paper of Aupetit and Mouton (1996), we consider the spectral definitions of rank, trace and determinant applied to elements in a general Banach algebra. We prove a generalization of Sylvester's Determinant Theorem to Banach algebras and thereafter a generalization of the Frobenius inequality.

Ranked solutions of the matric equation $A_{1} X_{1} = A_{2} X_{2}$ .

Porter, A.Duane, Mousouris, Nick (1980)

International Journal of Mathematics and Mathematical Sciences

Rank-one LMI approach to robust stability of polynomial matrices

Didier Henrion, Kenji Sugimoto, Michael Šebek (2002)

Kybernetika

Necessary and sufficient conditions are formulated for checking robust stability of an uncertain polynomial matrix. Various stability regions and uncertainty models are handled in a unified way. The conditions, stemming from a general optimization methodology similar to the one used in $μ$ -analysis, are expressed as a rank-one LMI, a non-convex problem frequently arising in robust control. Convex relaxations of the problem yield tractable sufficient LMI conditions for robust stability of uncertain...

Ranks and independence of solutions of the matrix equation $A X B + C Y D = M$ .

Tian, Yongge (2006)

Acta Mathematica Universitatis Comenianae. New Series

Ranks of permutative matrices

Xiaonan Hu, Charles R. Johnson, Caroline E. Davis, Yimeng Zhang (2016)

Special Matrices

A new type of matrix, termed permutative, is defined and motivated herein. The focus is upon identifying circumstances under which square permutative matrices are rank deficient. Two distinct ways, along with variants upon them are given. These are a special kind of grouping of rows and a type of partition in which the blocks are again permutative. Other, results are given, along with some questions and conjectures.

Rate of convergence to the semi-circle law for the Deformed Gaussian Unitary Ensemble

Friedrich Götze, Alexander Tikhomirov, Dmitry Timushev (2007)

Open Mathematics

It is shown that the Kolmogorov distance between the expected spectral distribution function of an n × n matrix from the Deformed Gaussian Ensemble and the distribution function of the semi-circle law is of order O(n −2/3+v ).

Rational algebra and MM functions

Ray A. Cuninghame-Green (2003)

Kybernetika

MM functions, formed by finite composition of the operators min, max and translation, represent discrete-event systems involving disjunction, conjunction and delay. The paper shows how they may be formulated as homogeneous rational algebraic functions of degree one, over (max, +) algebra, and reviews the properties of such homogeneous functions, illustrated by some orbit-stability problems.

Rational orthogonal versus real orthogonal.

Đoković, Dragomir Ž., Severini, Simone, Szöllősi, Ferenc (2009)

ELA. The Electronic Journal of Linear Algebra [electronic only]

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