on the inverse eigenvalue problem for nonnegative matrices of order two to five abstract in this paper for a given set of numbers with special conditions we construct a nonnegative matrix such that the given set is its spectrum

performance advantage of complex lms for controlling narrow band adaptive arrays in narrow band adaptive array applications the mean square convergence of the discrete time real least mean square lms algorithm is slowed by image frequency noises generated in the lms loops the complex lms algorithm proposed by widrow et al is shown to eliminate these noises yielding convergence of the mean squared error mse at slightly over twice the rate this paper includes a comprehensive analysis of the mse of adaptation for lms the analysis is based upon the method developed in the 1968 dissertation by k d senne and it represents the most complete treatment of the subject published to date

the inverse eigenvalue problem for nonnegative matrices abstract let a be a nonnegative matrix with spectrum 1 2 m and b be a nonnegative matrix with spectrum 1 2 n where 1 is the perron eigenvalue of b furthermore let a maximal diagonal element of a be greater than or equal to 1 in the article we construct a nonnegative matrix c with spectrum 1 2 m 2 n this construction enables us to obtain several results on how to determine new realizable lists from known realizable lists

nonnegative realizability with jordan structure abstract a general method is given for merging blocks in the jordan canonical form of a nonnegative matrix as a consequence results more general than any prior ones are given for the universal realizability of spectra that is spectra which are realizable by a nonnegative matrix for each possible jordan canonical form allowed by the spectrum in particular we generalize a classical result due to minc regarding positive diagonalizable matrices for example any spectrum that is diagonalizably realizable by a nonnegative matrix with mostly positive off diagonal entries is universally realizable

adaptive nonlinear system identification in the short time fourier transform domain in this paper we introduce an adaptive algorithm for nonlinear system identification in the short time fourier transform stft domain the adaptive scheme consists of a parallel combination of a linear component represented by crossband filters between subbands and a quadratic component which is modeled by multiplicative cross terms we adaptively update the model parameters using the least mean square lms algorithm and derive explicit expressions for the transient and steady state mean square error mse in frequency bins for white gaussian inputs we show that estimation of the nonlinear component improves the mse performance only when the power ratio of nonlinear to linear components is relatively high furthermore as the number of crossband filters increases a lower steady state mse may be obtained at the expense of slower convergence experimental results support the theoretical derivations

negativity compensation in the nonnegative inverse eigenvalue problem abstract if a set of complex numbers can be partitioned as 1 s in such a way that each i is realized as the spectrum of a nonnegative matrix say a i then is trivially realized as the spectrum of the nonnegative matrix a a i in linear algebra appl 369 2003 169 it was shown that in some cases a real set can be realized even if some of the i are not realizable themselves here we systematize and extend these results in particular allowing the sets to be complex the leading idea is that one can associate to any nonrealizable set a certain negativity n and to any realizable set a certain positivity m then under appropriate conditions if m n we can conclude that is the spectrum of a nonnegative matrix additionally we prove a complex generalization of suleimanova s theorem

nonnegative matrices with prescribed spectrum and elementary divisors abstract in this paper we give new sufficient conditions for the existence and construction of nonnegative matrices with prescribed elementary divisors which drastically improve and contain some of the previous known conditions we also show how to perturb complex eigenvalues of a nonnegative matrix while keeping its nonnegativity these results allow us under certain conditions to easily decide if a given list is realizable with prescribed elementary divisors

the factorizations of the finite exceptional groups of lie type a finite group is said to be factorizable if g ab for some proper subgroups a and b of g such an expression is called a factorization of g factorizations have been the subject of considerable study see for example 11 121 for the case where g is a finite simple group some results can be found in 8 20 26 27 291 in this paper we determine completely the factorizations of the exceptional simple groups of lie type and their automorphism groups here by an exceptional simple group of lie type we mean a non abelian finite simple group associated with one of the families g f4 e e e b 2g2 3d f4 and 2e excluding g 2 and g 3 in view of the isomorphisms g 2 e u 3 and g 3 z l 8 there are two interesting features of our main result one is that remarkably few of the exceptional simple groups of lie type are factorizable for example e q ec q e q and e q possess no factorizations in contrast some striking factorizations are found to exist for the groups g q and f q for g q these factorizations are already known 20 271 in another paper 151 the maximal factorizations of all the other finite simple groups and their automorphism groups are classified where the factorization g ab is maximal if both a and b are maximal subgroups of g the results of this paper and 151 are used in 161 to obtain a classification of the maximal subgroups of the finite alternating and symmetric groups our main results here are the following two theorems

inverse maximal eigenvalues problems for leslie and doubly leslie matrices abstract in this paper we deal with leslie and doubly leslie matrices of order n in particular with the companion and doubly companion matrices we study three inverse eigenvalues problems which consist of constructing these matrices from the maximal eigenvalues of its all leading principal submatrices for leslie and doubly companion matrices an eigenvector associated with the maximal eigenvalue of the matrix is additionally considered and for the doubly leslie matrix also an eigenvector associated with the maximal eigenvalue of leading principal submatrix of order n 1 is required we give necessary and sufficient conditions for the existence of a leslie matrix and a companion matrix and sufficient conditions for the existence of a doubly leslie matrix and a doubly companion matrix our results are constructive and generate an algorithmic procedure to construct these special kinds of matrices

steady state performance of spline adaptive filters recently a novel class of nonlinear adaptive filters called spline adaptive filters safs has been introduced and demonstrated to be very effective in many practical applications the learning rules of these architectures are based on the least mean square lms algorithm in order to provide theoretical foundation to the saf in this paper we provide a steady state performance evaluation in particular after the stochastic analysis of the mean behavior of the saf approach under the gaussian assumption the analytical derivation of the theoretical excess mean square error emse and the normalized misadjustment are derived and discussed the proposed analysis of emse and misadjustment is based on the energy conservation approach that has been extended to saf architecture the derived theoretical analysis allows to accurately predict the steady state performance therefore some properties for the correct choice of filter parameters are also provided experimental results demonstrate the effectiveness of the analysis results

matrix completion problems over integral domains the case with a diagonal of prescribed blocks abstract let r be an arbitrary integral domain let 1 n be a multiset of elements of r let be a permutation of 1 k let n 1 n k be positive integers such that n 1 n k n and for r 1 k let a r r n r n r we are interested in the problem of finding a block matrix q q rs r s 1 k r n n with spectrum and such that q r r a r for r 1 k cravo and silva completely characterized the existence of such a matrix when r is a field in this work we construct a solution matrix q that solves the problem when r is an integral domain with two exceptions i k 2 ii k 3 r r and n r n 2 for some r what makes this work quite unique in this area is that we consider the problem over the more general algebraic structure of integral domains which includes the important case of integers furthermore we provide an explicit and easy to implement finite step algorithm that constructs an specific solution matrix we point out that cravo and silva s proof is not constructive

a matrix completion problem over integral domains the case with 2n 3 prescribed entries abstract let 1 n n 2 be a given multiset of elements in an integral domain r and let p be a matrix of order n with at most 2 n 3 prescribed entries that belong to r under the assumption that each row each column and the diagonal of p have at least one unprescribed entry we prove that p can be completed over r to obtain a matrix a with spectrum we describe an algorithm to construct a this result is an extension to integral domains of a classical completion result by herskowitz for fields

universal realizability of spectra with two positive eigenvalues abstract a list of eigenvalues is said to be realizable if it is the spectrum of a nonnegative matrix diagonalizably realizable dr if it is the spectrum of a diagonalizable nonnegative matrix and universally realizable ur if there is a nonnegative matrix with this spectrum and any possible jordan canonical form allowed by the spectrum the nonnegative inverse eigenvalue problem niep asks which lists are realizable it is known that there exist spectra that are realizable but not dr we raise the question of whether dr implies ur this is known in a few cases including n 4 nonnegative spectra and suleimanova spectra we add some new classes of spectra that are ur

a nordhaus gaddum conjecture for the minimum number of distinct eigenvalues of a graph abstract we propose a nordhaus gaddum conjecture for q g the minimum number of distinct eigenvalues of a symmetric matrix corresponding to a graph g for every graph g excluding four exceptions we conjecture that q g q g c g 2 where g c is the complement of g we compute q g c for all trees and all graphs g with q g g 1 and hence we verify the conjecture for trees unicyclic graphs graphs with q g 4 and for graphs with g 7

constructing new realisable lists from old in the niep abstract given a list of complex numbers 1 2 n we say that is realisable if is the spectrum of some entrywise nonnegative matrix the nonnegative inverse eigenvalue problem or niep is the problem of characterising all realisable lists given a realisable list 2 3 m where is the perron eigenvalue and 2 is real we find families of lists 1 2 n for which 1 2 n 3 4 m is realisable in addition given a realisable list i i 4 5 m where is the perron eigenvalue and and are real we find families of lists 1 2 3 4 for which 1 2 3 4 4 5 m is realisable

elgenvalues of nonnegative matrices abstract in a partial continuation of work by fiedler some spectral properties of symmetric nonnegative matrices are extended to general nonnegative matrices

nonnegative realization of spectra having negative real parts abstract the nonnegative inverse eigenvalue problem niep is the problem of determining necessary and sufficient conditions for a list of complex numbers to be the spectrum of a nonnegative matrix in this paper the problem is completely solved in the case when all numbers in the given list except for one the perron eigenvalue have real parts smaller than or equal to zero

power series with positive coefficients arising from the characteristic polynomials of positive matrices let a be an n times n entrywise positive matrix and let f t det i t a we prove the surprising result that there always exists a positive integer n such that the formal power series expansion of 1 f t 1 n around t 0 has positive coefficients

variable length stochastic gradient algorithm the transversal variable length stochastic gradient algorithm is described it is a modification of the stochastic gradient algorithm that allows dynamic allocation of coefficients of an adaptive filter the order of the filter and the adaptation step size are changed automatically when an appropriate level of performance is reached during the course of the adaptation process in this way the algorithm results in fast convergence typical of low order filters and good steady state performance typical of high order filters

on nonnegative matrices with prescribed eigenvalues and diagonal entries abstract we consider the problem of the existence and construction of nonnegative matrices with prescribed spectrum and diagonal entries necessary and sufficient conditions have been obtained for n 3 by perfect and fiedler in the cases nonnegative and nonnegative symmetric respectively for n 4 they obtained sufficient conditions many partial results about the problem have been published by several authors mainly by smigoc this is a long standing unsolved inverse problem but also necessary to apply a perturbation result due to r rado which has played an important role in the study of nonnegative inverse eigenvalue and inverse elementary divisors problems distinct versions of rado s result have been also obtained for certain structured matrices to apply rado s result and its different versions we need to guarantee the existence of an r r r n nonnegative structured matrix with prescribed spectrum and diagonal entries this is an important motivation for this work here we prove new sufficient conditions for n 4 which extend and strictly contain the perfect and fiedler conditions our results generate an algorithmic procedure to construct a solution matrix

block matrices and guo s index for block circulant matrices with circulant blocks abstract in this paper we deal with circulant and partitioned into n by n circulant blocks matrices and introduce spectral results concerning this class of matrices the problem of finding lists of complex numbers corresponding to a set of eigenvalues of a nonnegative block matrix with circulant blocks is treated along the paper we call realizable list if its elements are the eigenvalues of a nonnegative matrix the guo s index 0 of a realizable list is the minimum spectral radius such that the list up to the initial spectral radius together with 0 is realizable the guo s index of block circulant matrices with circulant blocks is obtained and in consequence necessary and sufficient conditions concerning the niep nonnegative inverse eigenvalue problem for the realizability of some spectra are given

realizability criterion for the symmetric nonnegative inverse eigenvalue problem abstract let 1 2 n a set of real numbers the real nonnegative inverse eigenvalue problem rniep is the problem of determining necessary and sufficient conditions in order that be the spectrum of an entrywise nonnegative n n matrix if there exists a nonnegative matrix a with spectrum we say that is realized by a many realizability criteria for the existence of such a matrix a are known this paper shows that a realizability criterion given by the author which contains both kellogg s realizability criterion and borobia s realizability criterion is sufficient for the existence of an n n symmetric nonnegative matrix with prescribed spectrum

a characterization of trace zero nonnegative 5 5 matrices let 1 2 3 4 5 be a list of complex numbers with 1 2 3 4 5 0 necessary and sufficient conditions for the existence of an entry wise nonnegative 5 5 matrix a with spectrum are presented

a note on the convergence analysis of lms adaptive filters with gaussian data necessary and sufficient conditions for the convergence of lms least mean squares adaptive filters with gaussian data have been established by l l horowitz and k d senne 1981 with the recent support of a feuer and e weinstein 1985 a feature of both of these studies is the necessity to investigate bounds on the roots of rather unwieldy characteristic equations the authors show how such an investigation can be avoided through the use of a theorem of f r gantmacher 1959 in formally applying this theorem similar results to those of the above studies are obtained in a precise and straightforward manner

realizable lists on a class of nonnegative matrices abstract a square matrix of order n with n 2 is called permutative matrix when all its rows are permutations of the first row in this paper recalling spectral results for partitioned into 2 by 2 symmetric blocks matrices sufficient conditions on a given complex list to be the list of the eigenvalues of a nonnegative permutative matrix are given in particular we study niep and pniep when some complex elements in the lists under consideration have non zero imaginary part realizability regions for nonnegative permutative matrices are obtained a guo s realizability preserving perturbations result is obtained

realizable lists via the spectra of structured matrices abstract a square matrix of order n with n 2 is called a permutative matrix or permutative when all its rows up to the first one are permutations of precisely its first row in this paper the spectra of a class of permutative matrices are studied in particular spectral results for matrices partitioned into 2 by 2 symmetric blocks are presented and using these results sufficient conditions on a given list to be the list of eigenvalues of a nonnegative permutative matrix are obtained and the corresponding permutative matrices are constructed guo perturbations on given lists are exhibited

an algorithm for constructing nonnegative matrices with prescribed real eigenvalues provided with the real spectrum this paper presents a numerical procedure based on the induction principle to solve two types of inverse eigenvalue problems one for nonnegative matrices and another for symmetric nonnegative matrices as an immediate application our approach can offer not only the sufficient condition to solve inverse eigenvalue problems for nonnegative or symmetric nonnegative matrices but also a numerical way to solve inverse eigenvalue problems for stochastic matrices numerical examples are presented to show the capacity of our method

perturbing non real eigenvalues of non negative real matrices abstract let b i c b i c 4 n be the spectrum of an entry non negative matrix and t 0 laffey t j laffey perturbing non real eigenvalues of nonnegative real matrices electron j linear algebra 12 2005 73 76 has shown that 2 t b t i c b t i c 4 n is also the spectrum of some nonnegative matrix laffey 2005 has used a rank one perturbation for small t and then used a compactness argument to extend the result to all nonnegative t in this paper a rank two perturbation is used to deduce an explicit and constructive proof for all t 0

steady state solution of the deficient length constrained fblms algorithm in almost all analyses of the frequency domain block least mean square fblms algorithm the length of the adaptive filter is assumed to be equal to that of the unknown system impulse response however in many practical situations where the length of the system impulse response is extremely long the adaptive filter usually works in an under modeling situation the analysis results for the sufficient length fblms algorithm are not applicable to the deficient length case in this correspondence the steady state solution of the deficient length constrained fblms algorithm is analyzed simulations illustrate the accuracy of the theoretical results

the nonnegative inverse eigenvalue problem abstract let 1 n be a list of complex numbers the nonnegative inverse eigenvalue problem niep is the problem of determining necessary and sufficient conditions in order that be the spectrum of an entrywise nonnegative n n matrix our purpose is to give an overview of the niep its history modern results and subproblems with particular emphasis on symmetric realizing matrices a substantial bibliography is included

strong shift equivalence and the generalized spectral conjecture for nonnegative matrices abstract given matrices a and b shift equivalent over a dense subring r of r with a primitive we show that b is strong shift equivalent over r to a primitive matrix this result shows that the weak form of the generalized spectral conjecture for primitive matrices implies the strong form the foundation of this work is the recent result that for any ring r the group nk 1 r of algebraic k theory classifies the refinement of shift equivalence by strong shift equivalence for matrices over r

on the learning mechanism of adaptive filters this paper highlights both analytically and by simulations some interesting phenomena regarding the behavior of ensemble average learning curves of adaptive filters that may have gone unnoticed among other results the paper shows that even ensemble average learning curves of single tap lms filters actually exhibit two distinct rates of convergence one for the initial time instants and another faster one for later time instants in addition such curves tend to converge faster than predicted by mean square theory and can converge even when a mean square stability analysis predicts divergence these effects tend to be magnified by increasing the step size two of the conclusions that follow from this work are 1 the mean square stability alone may not be the most appropriate performance measure especially for larger step sizes a combination of mean square stability and almost sure a s stability seems to be more appropriate 2 care is needed while interpreting ensemble average curves for larger step sizes the curves can lead to erroneous conclusions unless a large number of experiments are averaged at times of the order of tens of thousands or higher

quantifying the effects of dimension on the convergence rate of the lms adaptive fir estimator the convergence rate of an lms adaptive fir filter to an unknown stationary channel may be influenced by the filter parameter dimension as well as by the input signal s characteristics this dimension influence may be of importance in applications such as adaptive acoustic echo cancellation in which the unknown channel is typically modeled as a long fir filter the paper includes the development and proposal of a novel measure of the expected convergence rate of the lms fir filter followed by analysis of this convergence rate measure the analysis indicates that unless the input signal is white the expected convergence rate decreases with increasing dimension down to a limiting value which is determined by the input signal s autocorrelation level

the real nonnegative inverse eigenvalue problem is np hard abstract a list of complex numbers is realizable if it is the spectrum of a nonnegative matrix in 1949 suleimanova posed the nonnegative inverse eigenvalue problem niep the problem of determining which lists of complex numbers are realizable the version for reals of the niep rniep asks for realizable lists of real numbers in the literature there are many sufficient conditions or criteria for lists of real numbers to be realizable we will present an unified account of these criteria then we will see that the decision problem associated to the rniep is np hard and we will study the complexity for the decision problems associated to known criteria

on universal realizability of spectra abstract a list 1 2 n of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix the list is said to be universally realizable ur if it is the spectrum of a nonnegative matrix for each possible jordan canonical form allowed by it is well known that an n n nonnegative matrix a is co spectral to a nonnegative matrix b with constant row sums in this paper we extend the co spectrality between a and b to a similarity between a and b when the perron eigenvalue is simple we also show that if 0 and 1 2 n is ur then 1 2 n is also ur we give counter examples for the cases 1 2 n is ur implies 1 2 3 n is ur and 1 2 are ur implies 1 2 is ur

performance analysis of the deficient length lms adaptive algorithm in almost all analyzes of the least mean square lms finite impulse response fir adaptive algorithm it is assumed that the length of the adaptive filter is equal to that of the unknown system impulse response however in many practical situations a deficient length adaptive filter whose length is less than that of the unknown system is employed and analysis results for the sufficient length lms algorithm are not necessarily applicable to the deficient length case therefore there is an essential need to accurately quantify the behavior of the lms algorithm for realistic situations where the length of the adaptive filter is deficient in this paper we present a performance analysis for the deficient length lms adaptive algorithm for correlated gaussian input data and using the common independence assumption exact expressions that completely characterize the transient and steady state mean square performances of the algorithm are developed which lead to new insights into the statistical behavior of the deficient length lms algorithm simulation experiments illustrate the accuracy of the theoretical results in predicting the convergence behavior of the algorithm

the maximum of the minimal multiplicity of eigenvalues of symmetric matrices whose pattern is constrained by a graph abstract in this paper we introduce a parameter mm g defined as the maximum over the minimal multiplicities of eigenvalues among all symmetric matrices corresponding to a graph g we develop basic properties of mm g and compute it for several families of graphs

convergence analysis of lms filters with uncorrelated gaussian data statistical analysis of the least mean squares lms adaptive algorithm with uncorrelated gaussian data is presented exact analytical expressions for the steady state mean square error mse and the performance degradation due to weight vector misadjustment are derived necessary and sufficient conditions for the convergence of the algorithm to the optimal wiener solution within a finite variance are derived it is found that the adaptive coefficient which controls the rate of convergence of the algorithm must be restricted to an interval significantly smaller than the domain commonly stated in the literature the outcome of this paper therefore places fundamental limitations on the mse performance and rate of convergence of the lms adaptive scheme

connecting sufficient conditions for the symmetric nonnegative inverse eigenvalue problem abstract we say that a list of real numbers is symmetrically realisable if it is the spectrum of some entrywise nonnegative symmetric matrix the symmetric nonnegative inverse eigenvalue problem sniep is the problem of characterising all symmetrically realisable lists in this paper we present a recursive method for constructing symmetrically realisable lists the properties of the realisable family we obtain allow us to make several novel connections between a number of sufficient conditions developed over forty years starting with the work of fiedler in 1974 we show that essentially all previously known sufficient conditions are either contained in or equivalent to the family we are introducing

atlas of finite groups maximal subgroups and ordinary characters for simple groups this atlas covers groups from the families of the classification of finite simple groups recently updated incorporating corrections

performance analysis of deficient length quaternion least mean square adaptive filters quaternion adaptive filters have been widely used for processing 3d and 4d phenomena deficient length quaternion adaptive filters are explicitly or implicitly used in many practical applications where the length of system impulse response is large or unknown however their statistical behaviors are yet to be fully understood as theoretical results on the class of full length quaternion least mean square qlms algorithms do not necessarily apply to their deficient length versions this article fills this void and analyses the mean and mean square convergence of the deficient length qlms algorithms both for the strictly linear and widely linear cases transient and steady state performance is characterised by exploiting the augmented statistics of noncircular quaternion random vectors a novel decorrelation technique in the quaternion domain is shown to allow for the development of intuitive closed form solutions for correlated quaternion gaussian inputs thus unveiling the relationship between the algorithm behaviour and the noncircularity of quaternion input data the analysis also provides a general framework whereby the strictly linear and semi widely linear qlms algorithms can be seen as deficient length versions of the widely linear qlms numerical simulations validate the accuracy of the theoretical results and support the behaviour of the considered algorithms

spectra of principal submatrices of nonnegative matrices abstract let a be an n n nonnegative matrix with the spectrum 1 2 n and let a 1 be an m m principal submatrix of a with the spectrum 1 2 m in this paper we present some cases where the realizability of 1 2 m 1 2 s implies the realizability of 1 2 n 1 2 s and consider the question whether this holds in general in particular we show that the list 1 2 n 1 2 m is realizable

a unified approach to the steady state and tracking analyses of adaptive filters most adaptive filters are inherently nonlinear and time variant systems the nonlinearities in the update equations tend to lead to difficulties in the study of their steady state performance as a limiting case of their transient performance this paper develops a unified approach to the steady state and tracking analyses of adaptive algorithms that bypasses many of these difficulties the approach is based on the study of the energy flow through each iteration of an adaptive filter and it relies on a fundamental error variance relation

a full mean square analysis of clms for second order noncircular inputs a full mean square transient and steady state analysis of the complex least mean square clms algorithm is provided for strictly linear estimation of general second order noncircular improper gaussian inputs to this end we also consider the performance assessment in terms of the evolution of the complementary mean square error cmse and the complementary covariance pseudocovariance matrix of the weight error vector of clms this makes it possible to measure the degrees of noncircularity of the output error and the weight error vector which arise due to second order noncircularity improperness of the system input and system noise the recently introduced approximate uncorrelating transform which allows for joint direct diagonalization of both the input covariance and complementary covariance matrices with a single singular value decomposition is then employed in order to derive a unified bound on the step size which guarantees the convergence of both the standard mse and the proposed cmse a joint consideration of the standard mean square performance analysis and the proposed complementary performance analysis is shown to provide full second order closed form statistical descriptions of both the transient and steady state performances of clms for second order noncircular improper gaussian input data simulations in the system identification setting support the analysis

graphs that allow all the eigenvalue multiplicities to be even abstract let g be an undirected graph on n vertices and let s g be the set of all n n real symmetric matrices whose nonzero off diagonal entries occur in exactly the positions corresponding to the edges of g the inverse eigenvalue problem for a graph g is a problem of determining all possible lists that can occur as the lists of eigenvalues of matrices in s g this question is in general hard to answer and several variations were studied most notably the minimum rank problem in this paper we introduce the problem of determining for which graphs g there exists a matrix in s g whose characteristic polynomial is a square i e the multiplicities of all its eigenvalues are even we solve this question for several families of graphs

error energy bounds for adaptive gradient algorithms the paper establishes robustness optimality and convergence properties of the widely used class of instantaneous gradient adaptive algorithms the analysis is carried out in a purely deterministic framework and assumes no a priori statistical information it employs the cauchy schwarz inequality for vectors in an euclidean space and derives local and global error energy bounds that are shown to highlight as well as explain relevant aspects of the robust performance of adaptive gradient filters along the lines of h sup spl infin theory

perturbations on constructible lists preserving realizability in the niep and questions of monov abstract in 5 the authors showed that if 1 2 2 4 n is realizable where 1 is the perron eigenvalue and 2 is non real then so too is 1 4 t 2 t 2 t 4 n they asked if 4 can be replaced by 1 or 2 or what is the least possible multiple c 1 of t in order for this perturbation to be realizable in 2 the authors showed that for n 4 one can find certain spectra for which the result holds when c 1 provided t is small in this paper we show that c 2 is best possible and we construct a realizing matrix for c 2 when t is sufficiently small we also address some questions of monov concerning the realizability of the derivative of a realizable polynomial and if such a polynomial must have positive power sums

extreme nonnegative matrices abstract an entrywise nonnegative n n matrix a is extreme if its spectrum 1 n has the property that for all 0 1 n is not the spectrum of a nonnegative matrix it is proved that if a is an extreme nonnegative matrix then there exists a nonzero nonnegative matrix y such that ay ya and at y 0 where is the hadamard or entrywise product and t denotes transpose

construction of nonnegative symmetric matrices with given spectrum abstract let 1 n be the spectrum of a nonnegative symmetric matrix a with the perron eigenvalue 1 a diagonal entry c and let 1 m be the spectrum of a nonnegative symmetric matrix b with the perron eigenvalue 1 we show how to construct a nonnegative symmetric matrix c with the spectrum 1 max 0 1 c 2 n 2 m

applications of a brauer theorem in the nonnegative inverse eigenvalue problem abstract a result by brauer which shows how to modify one single eigenvalue of a matrix without changing any of the remaining eigenvalues plays a relevant role in the study of the nonnegative inverse eigenvalue problem niep perfect in a long time ignored paper 1955 presents an extension of this result which shows how to modify r eigenvalues of a matrix of order n r n via a rank r perturbation without changing any of the n r remaining eigenvalues by using this extension perfect gives a realizability criterion for the real niep which is not contained in soto s realizability criterion in this work by extending perfect s result we give a new realizability criterion for the real niep which contains soto s criterion thus this new realizability criterion appears to be the most general sufficient condition for the real niep so far we also contribute to the solution of the symmetric niep for n 5

perturbing eigenvalues of nonnegative matrices abstract let a be an irreducible entrywise nonnegative n n matrix with eigenvalues 2 b i c 3 b i c 4 n where is the perron eigenvalue it is shown that for any t 0 there is a nonnegative matrix with eigenvalues t 2 t 3 t 4 n whenever t n t with 3 1 4 2 5 5 and n 2 25 for n 6 the result improves that of guo et al our proof depends on an auxiliary result in geometry asserting that the area of an n sided convex polygon is bounded by n times the maximum area of a triangle lying inside the polygon

an inequality for the spectra of nonnegative matrices abstract let a be a n n entrywise nonnegative matrix and let s k trace a k k 1 2 3 it is shown that if n 1 then n 2 s 3 3 ns 1 s 2 2 s 1 3 n 2 n 1 ns 2 s 1 2 3 2 is nonnegative the result is used to show that if 1 2 2 4 n is the spectrum of a nonnegative matrix where 2 is nonreal and 1 max j j 1 n then 1 t 2 t 2 t 4 n need not be realizable for all t 0 even when re 2 0

guo perturbation for symmetric nonnegative circulant matrices abstract guo w guo eigenvalues of nonnegative matrices linear algebra appl 266 1997 261 270 sets the question if the list 1 2 n is symmetrically realizable that is is the spectrum of a symmetric nonnegative matrix and t 0 whether or not the list t 1 t 2 t 3 n is also symmetrically realizable in this paper we give an affirmative answer to this question in the case that the realizing matrix is circulant or left circulant we also give a necessary and sufficient condition for to be the spectrum of a nonnegative left circulant matrix

on the spectra of some g circulant matrices and applications to nonnegative inverse eigenvalue problem abstract a g circulant matrix a is defined as a matrix of order n where the elements of each row of a are identical to those of the previous row but are moved g positions to the right and wrapped around using number theory certain spectra of g circulant real matrices are given explicitly the obtained results are applied to nonnegative inverse eigenvalue problem to construct nonnegative g circulant matrices with given appropriated spectrum additionally some g circulant matrices are reconstructed from its main diagonal entries

factorizations of almost simple groups with a factor having many nonsolvable composition factors abstract this paper classifies the factorizations of almost simple groups with a factor having at least two nonsolvable composition factors this together with a previous classification result of the authors reduces the factorization problem of almost simple groups to the case where both factors have a unique nonsolvable composition factor