New light on the theorem of Perron.
We prove that the principal eigenvector of a positive matrix represents the relative dominance of its rows or ranking of alternatives in a decision represented by the rows of a pairwise comparison matrix.
We prove that the principal eigenvector of a positive matrix represents the relative dominance of its rows or ranking of alternatives in a decision represented by the rows of a pairwise comparison matrix.
Let ⊞, ⊠, and ⊎ be the free additive, free multiplicative, and boolean additive convolutions, respectively. For a probability measure μ on [0,∞) with finite second moment, we find a scaling limit of as N goes to infinity. The -transform of its limit distribution can be represented by Lambert’s W-function. From this, we deduce that the limiting distribution is freely infinitely divisible, like the lognormal distribution in the classical case. We also show a similar limit theorem by replacing free...
Our purpose is to present a number of new facts about the structure of semipositive matrices, involving patterns, spectra and Jordon form, sums and products, and matrix equivalence, etc. Techniques used to obtain the results may be of independent interest. Examples include: any matrix with at least two columns is a sum, and any matrix with at least two rows, a product, of semipositive matrices. Any spectrum of a real matrix with at least elements is the spectrum of a square semipositive matrix,...
In this paper we study -EP matrices, as a generalization of EP-matrices in indefinite inner product spaces, with respect to indefinite matrix product. We give some properties concerning EP and -EP matrices and find connection between them. Also, we present some results for reverse order law for Moore-Penrose inverse in indefinite setting. Finally, we deal with the star partial ordering and improve some results given in the “EP matrices in indefinite inner product spaces” (2012), by relaxing some...
We present new iterative methods for solving the Sylvester equation belonging to the class of SOR-like methods, based on the SOR (Successive Over-Relaxation) method for solving linear systems. We discuss convergence characteristics of the methods. Numerical experimentation results are included, illustrating the theoretical results and some other noteworthy properties of the Methods.
New necessary and sufficient conditions for asymptotic stability of positive continuous-discrete 2D linear systems are established. Necessary conditions for the stability are also given. The stability tests are demonstrated on numerical examples.
Soit un nombre de Pisot ; nous montrons que pour tout entier assez grand il existe une matrice carrée à coefficients positifs ou nuls dont l’ordre est égal au degré de et dont est valeur propre.Soit le -développement de ; si est un nombre de Pisot, alors la suite est périodique après un certain rang (pour , ) et le polynômeest appelé polynôme de Parry. Nous montrons qu’il existe un ensemble relativement dense d’entiers tels que le polynôme minimal de est égal à son polynôme...
Let , and be fixed complex numbers. Let be the Toeplitz matrix all of whose entries above the diagonal are , all of whose entries below the diagonal are , and all of whose entries on the diagonal are . For , each principal minor of has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of . We also show that all complex polynomials in are Toeplitz matrices. In particular, the inverse of is a Toeplitz matrix when...
A new formula is established for the asymptotic expansion of a matrix integral with values in a finite-dimensional von Neumann algebra in terms of graphs on surfaces which are orientable or non-orientable.