characteristic polynomials and controlability of partially prescribed matrices abstract in a previous paper it was proved that n 1 arbitrary entries and the characteristic polynomial of a n n matrix over a field f can be arbitrarily prescribed except if all the nonprincipal entries of a row or column are prescribed equal to zero and the characteristic polynomial does not have a root in f this paper describes the possible characteristic polynomials of a pk pk matrix partitioned into k k blocks of size p p when k 1 blocks are fixed and the others vary it also studies the possibility of a pair of matrices a 1 a 2 where a 1 is square and a 1 a 2 is partitioned into k k 1 blocks of size p p being completely controllable when some of the blocks are prescribed and the others vary

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

possible spectra of totally positive matrices abstract for any given set s of n distinct positive numbers we construct a symmetric n by n strictly totally positive matrix whose spectrum is s thus in order to be the spectrum of an n by n totally positive matrix it is necessary and sufficient that n numbers be positive and distinct

the rationality problem for finite subgroups of gl4 q abstract let g be a finite subgroup of gl 4 q the group g induces an action on q x 1 x 2 x 3 x 4 the rational function field of four variables over q theorem the fixed subfield q x 1 x 2 x 3 x 4 g f q x 1 x 2 x 3 x 4 f f for any g is rational i e purely transcendental over q except for two groups which are images of faithful representations of c 8 and c 3 c 8 into gl 4 q both fixed fields for these two exceptional cases are not rational over q there are precisely 227 such groups in gl 4 q up to conjugation the answers to the rationality problem for most of them were proved by kitayama and yamasaki 2009 ky except for four cases we solve these four cases left unsettled by kitayama and yamasaki thus the whole problem is solved completely

matrices with prescribed eigenvalues and blocks abstract we solve the following problem proposed by oliveira under what conditions does there exist a square matrix a a ij i j 1 2 a 11 is principal over an arbitrary field with prescribed eigenvalues and prescribed submatrices a 11 a 12 and a 22

the number of invariant polynomials of a matrix with prescribed off diagonal blocks abstract let a be an n n matrix over an arbitrary field f of the form a a 1 2 a 1 2 a 2 1 a 2 2 where a1 1 fp p and a2 2 fq q and p q n we characterize the possible number of nontrivial invariant polynomials of a when the submatrices a1 2 and a2 1 are prescribed

matrices with prescribed eigenvalues and principal submatrices abstract we give a necessary and sufficient condition for the existence of a square matrix with prescribed eigenvalues and prescribed complementary principal blocks

counting zero kernel pairs over a finite field abstract helmke et al have recently given a formula for the number of reachable pairs of matrices over a finite field we give a new and elementary proof of the same formula by solving the equivalent problem of determining the number of so called zero kernel pairs over a finite field we show that the problem is equivalent to certain other enumeration problems and outline a connection with some recent results of guo and yang on the natural density of rectangular unimodular matrices over f q x we also propose a new conjecture on the density of unimodular matrix polynomials

interlacing inequalities and control theory abstract given a p by p matrix a we solve the problem of the existence of a p by q matrix b such that a b has prescribed controllability indices and i p a b has prescribed invariant polynomials the solution of this problem together with an earlier theorem of the author s is used to provide a new proof of the sa thompson interlacing inequalities for invariant polynomials

matrices with prescribed eigenvalues and prescribed submatrices ii let f be a field and let n p 1 p 2 p 3 be positive integers such that n p 1 p 2 p 3 let c c 1 1 c 1 2 c 1 3 c 2 1 c 2 2 c 2 3 c 3 1 c 3 2 c 3 3 f n i n where the blocks c i j f p i i p j i j 1 2 3 in this paper we describe the eigenvalues of c when c 1 2 c 1 3 c 2 2 are prescribed with c 1 3 0 and the other blocks are unknown for the same prescription of blocks for arbitrary prescription of c 1 3 we still provide a sufficient condition for the prescription of the characteristic polynomial of c

scaling in non stationary time series i abstract most data processing techniques applied to biomedical and sociological time series are only valid for random fluctuations that are stationary in time unfortunately these data are often non stationary and the use of techniques of analysis resting on the stationary assumption can produce a wrong information on the scaling and so on the complexity of the process under study herein we test and compare two techniques for removing the non stationary influences from computer generated time series consisting of the superposition of a slow signal and a random fluctuation the former is based on the method of wavelet decomposition and the latter is a proposal of this paper denoted by us as step detrending technique we focus our attention on two cases when the slow signal is a periodic function mimicking the influence of seasons and when it is an aperiodic signal mimicking the influence of a population change increase or decrease for the purpose of computational simplicity the random fluctuation is taken to be uncorrelated however the detrending techniques here illustrated work also in the case when the random component is correlated this expectation is fully confirmed by the sociological applications made in the companion paper we also illustrate a new procedure to assess the existence of a genuine scaling based on the adoption of diffusion entropy multiscaling analysis and the direct assessment of scaling using artificial sequences we show that the joint use of all these techniques yield the detection of the real scaling and that this is independent of the technique used to detrend the original signal

the number of linear transformations defined on a subspace with given invariant factors abstract given a finite dimensional vector space v over the finite field f q and a subspace w of v we consider the problem of counting linear transformations t w v which have prescribed invariant factors the case w v is a well studied problem that is essentially equivalent to counting the number of square matrices over f q in a conjugacy class and an explicit formula is known in this case on the other hand the case of general w is also an interesting problem and there hasn t been substantive progress in this case for over two decades barring a special case where all the invariant factors of t are of degree zero we extend this result to the case of arbitrary w by giving an explicit counting formula as an application of our results we give new proofs of some recent enumerative results in linear control theory and derive an extension of the gerstenhaber reiner formula for the number of square matrices over f q with given characteristic polynomial

the number of nonconstant invariant polynomials of matrices with several prescribed blocks abstract consider a n n matrix partitioned into k k blocks c c i j where c 1 1 c k k are square this paper studies the possible numbers of nonconstant invariant polynomials of c when a diagonal of blocks c i j is fixed and the others vary

existence of matrices with prescribed entries abstract it is proved that apart from for some exceptional cases there always exists an n n nonderogatory matrix over an arbitrary field with n prescribed entries and prescribed characteristic polynomial

on the number of invariant polynomials of partially prescribed matrices abstract we study the number of invariant polynomials of a i j i j 1 2 when the submatrices a 1 1 a 1 2 a 2 2 are prescribed

the number of invariant polynomials of a matrix with prescribed complementary principal submatrices abstract we study the number of invariant polynomials of a i j i j 1 2 when the submatrices a 1 1 and a 2 2 are prescribed

matrices with prescribed rows and invariant factors abstract in this paper we solve the problem of the existence of an n n matrix over an arbitrary field when its invariant polynomials and either some rows or columns are prescribed the solution is given in terms of invariant factor inequalities and of majorization inequalities involving controllability indices and the degrees of the invariant polynomials

some matrix completions over integral domains abstract we characterize 3 3 nilpotent matrices which are completions of 2 2 arbitrary matrices and 3 3 idempotent matrices which are completions of 2 2 arbitrary matrices over integral domains as an application we show that a nil clean element of a ring which belongs to a corner of the ring may not be nil clean in this corner

matrix completion problems abstract throughout the last decades several results have been published in the area of the so called matrix completion problems in this paper we survey several results in this field in particular we describe the possible eigenvalues the characteristic polynomial the invariant polynomials or the number of nontrivial invariant polynomials of a square matrix over a field when some of its entries are prescribed and the others vary finally we present our contribution generalizing some of the previous cases to an n n matrix partitioned into k k blocks with entries in a field when some of its blocks are prescribed and the others vary

the characteristic polynomial of a matrix with prescribed off diagonal blocks abstract we present new sufficient conditions for the existence of a matrix with prescribed complementary off diagonal blocks and characteristic polynomial

matrices with prescribed characteristic polynomial and principal blocks ii abstract in this paper we present some new sufficient conditions for the existence of a square matrix with prescribed characteristic polynomial and two complementary principal submatrices

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

eigenvalues of matrices with several prescribed blocks ii abstract a previous paper by d hershkowitz linear and multilinear algebra 14 1983 315 342 described the possible eigenvalues of an n n matrix when 2n 3 entries are fixed and the others vary this paper describes the possible eigenvalues of a pk pk matrix partitioned into k k blocks of size p p when 2k 3 blocks are fixed and the others vary