Finding All Essential Terms Of A Characteristic Maxpolynomial

Linear Algebra and its Applications

Recently, in a work that grew out of their exploration of interlacing polynomials, Marcus, Spielman and Srivastava [21] and Marcus [20] studied certain combinatorial polynomial convolutions. These convolutions preserve real-rootedness and capture expectations of characteristic polynomials of unitarily invariant random matrices, thus providing a link to free probability. We explore analogues of these types of convolutions in the setting of max-plus algebra. In this setting the max-permanent replaces the determinant, the maximum is the analogue of the expected value and real-rootedness is replaced by full canonical form. Our results resemble those of Marcus et al., however, in contrast to the classical setting we obtain an exact and simple description of all roots of the convolution of p(x) and q(x) in terms of the roots of p(x) and q(x).

Let a ⊕ b = max(a, b), a ⊗ b = a + b for a, b ∈ R := R ∪ {−∞}. By max-algebra we understand the analogue of linear algebra developed for the pair of operations (⊕, ⊗) extended to matrices and vectors. Max-algebra, which has been studied for more than 40 years, is an attractive way of describing a class of nonlinear problems appearing for instance in machinescheduling, information technology and discrete-event dynamic systems. This paper focuses on presenting a number of links between basic max-algebraic problems like systems of linear equations, eigenvalue-eigenvector problem, linear independence, regularity and characteristic polynomial on one hand and combinatorial or combinatorial optimisation problems on the other hand. This indicates that max-algebra may be regarded as a linear-algebraic encoding of a class of combinatorial problems. The paper is intended for wider readership including researchers not familiar with max-algebra.

Transactions of the American Mathematical Society, 2012

We study the behavior of max-algebraic powers of a reducible nonnegative matrix A ∈ R + n × n A\in \mathbb {R}_+^{n\times n} . We show that for t ≥ 3 n 2 t\geq 3n^2 , the powers A t A^t can be expanded in max-algebraic sums of terms of the form C S t R CS^tR , where C C and R R are extracted from columns and rows of certain Kleene stars, and S S is diagonally similar to a Boolean matrix. We study the properties of individual terms and show that all terms, for a given t ≥ 3 n 2 t\geq 3n^2 , can be found in O ( n 4 log ⁡ n ) O(n^4\log n) operations. We show that the powers have a well-defined ultimate behavior, where certain terms are totally or partially suppressed, thus leading to ultimate C S t R CS^tR terms and the corresponding ultimate expansion. We apply this expansion to the question whether { A t y , t ≥ 0 } \{A^ty,\; t\geq 0\} is ultimately linear periodic for each starting vector y y , showing that this question can also be answered in O ( n 4 log ⁡ n ) O(n^4\log n) time. W...

2009

For a general n×n real matrix A = (ai j), there exist standard O(n3) algorithms to find λ , x1, ...,xn such as max j=1,2,...,n (ai j + x j) = λ + xi for all i = 1,2, . . . ,n. It is known that λ is unique and equals to the maximum cycle mean of A = (ai j). Paper considers the case when A = (ai j) is an ε-triangular Toeplitz matrix, i.e. a diagonal-constant matrix with ai j ∈ R∪{−∞}, ai j = ε ≤−(n−1)max i≤ j |ai j| for i > j, ai j ∈ R for i ≤ j, and present algorithms to determine λ , x1, ...,xn in O(n2) time.

TURKISH JOURNAL OF MATHEMATICS, 2019

We consider new kinds of max and min matrices, [ a max(i,j) ] i,j≥1 and [ a min(i,j) ] i,j≥1 , as generalizations of the classical max and min matrices. Moreover, their reciprocal analogues for a given sequence {an} have been studied. We derive their LU and Cholesky decompositions and their inverse matrices as well as the LU-decompositions of their inverses. Some interesting corollaries will be presented.

Mathematical Programming, 2003

Linear Algebra and its Applications, 1999

Dedicated to Hans Schneider on the occasion of his 70th birthday.

2007

We consider the generalized eigenvalue problem

Linear Algebra and its Applications, 2013

The main purpose of this paper is to propose five programs in C ++ for matrix computations and solving recurrent equations systems with entries in max plus algebra. 1