triangularizing matrix polynomials for an algebraically closed field f we show that any matrix polynomial p lambda in f lambda nbym n le m can be reduced to triangular form preserving the degree and the finite and infinite elementary divisors we also characterize the real matrix polynomials that are triangularizable over the real numbers and show that those that are not triangularizable are quasi triangularizable with diagonal blocks of sizes 1 times 1 and 2 times 2 the proofs we present solve the structured inverse problem of building up triangular matrix polynomials starting from lists of elementary divisors

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

generalized christoffel darboux formula for classical skew orthogonal polynomials we show that skew orthogonal functions defined with respect to jacobi weight wa b x 1 x a 1 x b a b 1 including the limiting cases of laguerre wa x xa e x a 1 and gaussian weight satisfy three term recursion relation in the quaternion space from this we derive generalized christoffel darboux gcd formulae for kernel functions arising in the study of the corresponding orthogonal and symplectic ensembles of random 2n 2n matrices using the gcd formulae we calculate the level densities and prove that in the bulk of the spectrum under appropriate scaling the eigenvalue correlations are universal we also provide evidence to show that there exists a mapping between skew orthogonal functions arising in the study of orthogonal and symplectic ensembles of random matrices

explicit polar decomposition and a near characteristic polynomial the 2 2 case abstract explicit algebraic formulas for the polar decomposition of a nonsingular real 2 2 matrix a are given as well as a classification of all integer 2 2 matrices that admit a rational polar decomposition these formulas lead to a functional identity which is satisfied by all nonsingular real 2 2 matrices a as well as by exactly one type of exceptional matrix a n for each n 2

practical computation of matrix functions abstract several new representations for an analytic function f a of a complex matrix a and in particular for e at and a t are derived which also are numerically useful in that they avoid the computation of eigenvalues of a

bulk asymptotics of skew orthogonal polynomials for quartic double well potential and universality in the matrix model we derive bulk asymptotics of skew orthogonal polynomials m 1 4 defined with respect to the weight exp 2nv x v x gx4 4 tx2 2 g 0 and t 0 such that m n cr where cr is the critical value that separates skew orthogonal polynomials with two cuts from those with one cut simultaneously we derive asymptotics for the recursive coefficients of skew orthogonal polynomials the proof is based on obtaining a finite term recursion relation between skew orthogonal polynomials and orthogonal polynomials and using asymptotic results of orthogonal polynomials derived by bleher and its semiclassical asymptotics of orthogonal polynomials riemann hilbert problems and universality in the matrix model ann math 150 185 1999 finally we apply these asymptotic results of skew orthogonal polynomials and their recursion coefficients in the generalized christoffel darboux formula ghosh s generalized christoffel darboux formula for skew orthogona

efficient exhaustive listings of reversible one dimensional cellular automata this paper looks at an algebraic formulation of one dimensional cellular automata using the formulation connections to combinatorial structures and graph theory become clear strong results about uniqueness and isomorphism allows us to outline effective algorithms for the generation of exhaustive lists of reversible one dimensional cellular automata and to count the number of distinct examples that exist these algorithms use the orderly algorithm methods to avoid the pitfalls of brute force searches

fast and stable algorithms for computing the principal nth root of a complex matrix and the matrix sector function this paper presents rapidly convergent and more stable recursive algorithms for finding the principal nth root of a complex matrix and the associated matrix sector function the developed algorithms significantly improve the computational aspects of finding the principal nth root of a matrix and the matrix sector function thus the developed algorithms will enhance the capabilities of the existing computational algorithms such as the principal nth root algorithm the matrix sign algorithm and the matrix sector algorithm for developing applications to control system problems

skew orthogonal polynomials differential systems and random matrix theory we study skew orthogonal polynomials with respect to the weight function exp 2v x with v x 2dk 1 uk k xk u2d 0 d 0 a finite subsequence of such skew orthogonal polynomials arising in the study of orthogonal and symplectic ensembles of random matrices satisfies a system of differential difference deformation equation the vectors formed by such subsequence have the rank equal to the degree of the potential in the quaternion sense these solutions satisfy certain compatibility condition and hence admit a simultaneous fundamental system of solutions

roots of real matrices abstract an algorithm for computing the roots of a matrix with real elements and the real part of the eigenvalues not zero will be described the algorithm is based on the sign matrix given in an earlier paper and the projectors of the matrix

on the matrix equation f x a abstract this paper deals with the matrix equation f x a where a c n n is a given matrix and is a complex holomorphic function defined on an open subset d of c it furnishes information concerning the set s of all solutions x c n n of x a the subset sp s of all solutions that are polynomials in a and the subset sd s of all diagonalizable solutions a necessary and sufficient condition for s to be nonempty is established relations among all solutions of x a lying in the same similarity class are established in the particular case where a r n n d is symmetric with respect to the real line r c and z f z for every z d a necessary and sufficient condition for s r n n to be nonempty is established necessary and sufficient conditions for sp and sd to be empty finite infinite and commutative are established explicit computations of all the elements of sp and sd are furnished and the numbers of elements of these two sets are determined necessary and sufficient conditions for the relations s sp s sd or sp sd to be satisfied are established necessary and sufficient conditions for the existence of normal hermitian skew hermitian positive definite and unitary solutions are established it is shown that the set p a of all polynomials p c x such that x p a is a solution of x a depends only on the minimal polynomial of a and not on the size of a an example is furnished of a matrix equation x a that has no solution in f n n for any field f but has a solution in r n n when r is a certain ring these results are applied to the particular equations xm a ex a and the fixed point equation x x

jordan structures of upper equivalent matrices abstract jordan structures of strictly lower triangular completions of matrices over an algebraically closed field are studied we give in terms of majorization a sufficient condition for the existence of a completion which preserve the spectrum of the original matrix and has prescribed jordan structure the case of nilpotent matrix was considered previously by krupnik and leibman

skew orthogonal polynomials and random matrix ensembles there is considerable interest in understanding the relation between random matrix ensembles and quantum chaotic systems in the context of the universality of energy level correlations in this connection while gaussian ensembles of random matrices have been studied extensively not much is known about ensembles with non gaussian weight functions dyson has shown that the n level correlation functions can be expressed in terms of a kernel function involving orthogonal and skew orthogonal polynomials orthogonal for matrix ensembles with unitary invariance and skew orthogonal for ensembles with orthogonal and symplectic invariances we have obtained the following results 1 skew orthogonal polynomials of both types are derived for the jacobi class of weight functions including the limiting cases of associated laguerre and hermite or gaussian 2 matrix integral representations are given for the general weight functions 3 asymptotic forms of the polynomials are obtained rigorously for the jacobi class and in the form of an ansatz for the general case 4 for the three types of ensembles the asymptotic n level correlation functions with appropriate scaling are shown to be universal being independent of the weight function and location in the spectrum and identical with the well known gaussian results this provides a rigorous justification for the universality of the gaussian ensemble results observed in quantum chaotic systems as expected the level density is not universal

perturbation of linear control systems abstract we study the variation of the controllability indices and the jordan structure of a pair of matrices a b under small perturbations

cyclic dimensions kernel multiplicities and gohberg kaashoek numbers abstract two geometric characteristics namely cyclic dimensions and kernel multiplicities are introduced for a square matrix the connection between these characteristics and gohberg kaashoek numbers is studied on this basis two simple geometric proofs are given for the theorem about the change of the jordan structure of a given matrix under small perturbation

the change of the jordan structure under one row perturbations we study the variation of the jordan structure of a complex square matrix when we make small perturbations on the elements of one row

on the frechet derivative of matrix functions abstract in 1956 rinehart 4 discussed the derivatives of matrix functions by considering differences a e a for matrices e commuting with a in that case the derivative turned out to be a in this paper the case of noncommutative a and e is treated this leads to the frechet derivative of the matrix function an explicit integral representation is obtained using an approach that is similar to the one in 5 a finite sum in polynomials of a is obtained the coefficients may be computed recursively this is useful in computing the frechet derivative which is needed for newton s method to solve nonlinear matrix equations

infinitely many nonradial solutions of a euclidean scalar field equation abstract we study the equation u b x u x u x r n u h 1 r n the existence of a nonradial solution has been an open problem for some time even in the autonomous case under suitable hypotheses mainly on we find an unbounded sequence of nonradial solutions if n 4 or n 6 we also obtain infinitely many solutions if b and are not rotationally symmetric with respect to x but satisfy only a weaker symmetry

the change of the jordan structure of a mtrix under small perturbations abstract in this paper a complete description including multiplicity is given for the jordan structure of a matrix which is a small perturbation of a matrix of known jordan structure the problem solved here was solved independently as noted and the other solution has been published in english however the solutions are different the problem is important and it seems desirable to have this solution widely available

computation of the principal nth roots of complex matrices a new computational algorithm for finding the principal n th roots and associated roots of complex matrices is developed via a recursive solution of a block discrete state equation

an extension problem for the cr fractional laplacian we show that the conformally invariant fractional powers of the sub laplacian on the heisenberg group are given in terms of the scattering operator for an extension problem to the siegel upper halfspace remarkably this extension problem is different from the one studied among others by caffarelli and silvestre we also prove an energy identity that yields a sharp trace sobolev embedding

extraction of mth roots in matrix rings over fields abstract a matrix a m n f f an arbitrary field with characteristic p not necessarily positive is an m th power in m n f if and only if each of its p x primary components is where p x runs through the irreducible factors of the minimal or characteristic polynomial of a this paper establishes necessary and sufficient criteria for determining when such a p x primary matrix is an m th power the criteria fall into three cases 1 p x x if e 1 e 2 is the sequence of exponents of p x which form the elementary divisors of a extended by adding infinitely many 0 terms the criterion states that for all i 1 e i 1 m 1 e im 0 or 1 2 p x x m not divisible by p if is a root of the separable core q x of p x and e f then the multiplicity of each elementary divisor of a must be representable as a sum of integers not necessarily distinct each of which is the degree of some irreducible factor over e of the polynomial x m 3 p x x m a power of p in this case the criterion combines the criteria in the first two cases if e 1 e 2 is the sequence of exponents of p x which form the elementary divisors of a extended by adding infinitely many 0 terms then for all i 1 e i 1 m 1 e im 0 or 1 and the multiplicity of each e i must be representable as a sum of integers not necessarily distinct each of which is the degree of some irreducible factor over e of the polynomial x m

functions and jordan canonical forms of riordan matrices abstract this paper collects results about riordan arrays in the framework of matrix functions actually the following methodology applies to any square matrix m m with exactly one eigenvalue of algebraic multiplicity m n generalized lagrange bases are used to construct hermite polynomials that interpolate a family of functions moreover we show a parallel application of such functions via jordan canonical forms and case studies are given

spectral decomposition of a matrix using the generalized sign matrix an algorithm for spectral decomposition is presented which does not require knowledge of eigenvalues and eigenvectors a set of eigenprojectors are defined which covers the entire spectrum of a matrix and special attention is given to the projection on the zero eigenvalue some useful applications are discussed in the paper

canonical form associated with the problem of perturbation of one column of a controllable pair abstract let a b be a completely controllable matrix pair when we consider the problem of characterizing the controllability indices of all the matrix pairs obtained by small perturbations on one column of b a new equivalence relation arises in a natural way this equivalence relation is a kind of partial feedback equivalence as a consequence the controllability indices are invariant but they do not form a complete system of invariants we have found new invariants for this equivalence relation and we have obtained a canonical form in terms of these invariants

on the relation between orthogonal symplectic and unitary matrix ensembles for the unitary ensembles of n n hermitian matrices associated with a weight function w there is a kernel expressible in terms of the polynomials orthogonal with respect to the weight function which plays an important role for the orthogonal and symplectic ensembles of hermitian matrices there are 2 2 matrix kernels usually constructed using skew orthogonal polynomials which play an analogous role these matrix kernels are determined by their upper left hand entries we derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles we also show that whenever w w is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w w general formulas are obtained for these extra terms we do not use skew orthogonal polynomials in the derivations

newton s method for the matrix square root one approach to computing a square root of a matrix a is to apply newton s method to the quadratic matrix equation f x x2 a 0 two widely quoted matrix square root iterations obtained by rewriting this newton iteration are shown to have excellent mathematical convergence properties however by means of a perturbation analysis and supportive numerical examples it is shown that these simplified iterations are numerically unstable a further variant of newton s method for the matrix square root recently proposed in the literature is shown to be for practical purposes numerically stable

moser trudinger and beckner onofri s inequalities on the cr sphere we derive sharp moser trudinger inequalities on the cr sphere the rst type is in the adams form for powers of the sublaplacian and for general spectrally dened operators on the space of cr pluriharmonic functions we will then obtain the sharp beckner onofri inequality for crpluriharmonic functions on the sphere and as a consequence a sharp logarithmic hardy littlewood sobolev inequality in the form given by carlen and loss

structure preserving stratification of skew symmetric matrix polynomials we study how elementary divisors and minimal indices of a skew symmetric matrix polynomial of odd degree may change under small perturbations of the matrix coefficients we investigate these change

asymptotics of skew orthogonal polynomials exact integral expressions of the skew orthogonal polynomials involved in orthogonal 1 and symplectic 4 random matrix ensembles are obtained the even rank skew orthogonal polynomials are average characteristic polynomials of random matrices from there asymptotics of the skew orthogonal polynomials are derived

uniqueness of matrix mth root in this paper we talk about the uniqueness of matrix mth root of any nonsingular matrix what s more we extend a famous theorem

regularity estimates for the solution and the free boundary of the obstacle problem for the fractional laplacian we use a characterization of the fractional laplacian as a dirichlet to neumann operator for an appropriate differential equation to study its obstacle problem we write an equivalent characterization as a thin obstacle problem in this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary

computing real square roots of a real matrix bjorck and hammarling 1 describe a fast stable schur method for computing a square root x of a matrix a x2 a we present an extension of their method which enables real arithmetic to be used throughout when computing a real square root of a real matrix for a nonsingular real matrix a conditions are given for the existence of a real square root and for the existence of a real square root which is a polynomial in a the number of square roots of the latter type is determined the conditioning of matrix square roots is investigated and an algorithm is given for the computation of a well conditioned square root

the matrix sign decomposition and its relation to the polar decomposition the sign function of a square matrix was introduced by roberts in 1971 we show that it is useful to regard s sign a as being part of a matrix sign decomposition a sn where n a2 1 2 this decomposition leads to the new representation sign a a a2 1 2 most results for the matrix sign decomposition have a counterpart for the polar decomposition a uh and vice versa to illustrate this we derive best approximation properties of the factors u h and s determine bounds for a s and a u and describe integral formulas for s and u we also derive explicit expressions for the condition numbers of the factors s and n an important equation expresses the sign of a block 2 2 matrix involving a in terms of the polar factor u of a we apply this equation to a family of iterations for computing s by pandey kenney and laub to obtain a new family of iterations for computing u the iterations have some attractive properties including suitability for parallel computation

a parallel algorithm for principal n th roots of matrices an iterative algorithm for computing the principal nth root of a positive definite matrix is presented the algorithm is based on the gauss legendre approximation of a definite integral we present a parallelization in which we use as many processors as the order of the approximation an analysis of the error introduced at each step of the iteration indicates that the algorithm converges more rapidly as the order of the approximation and thus the number of processors increases we describe the results of our implementation of an eight processor meiko cs 2 comparing the parallel algorithm to the fastest sequential algorithm which is the hoskins walton method

a schur method for the square root of a matrix abstract a fast and stable method for computing the square root x of a given matrix a x 2 a is developed the method is based on the schur factorization a qsq h and uses a fast recursion to compute the upper triangular square root of s it is shown that if x 2 a is not large then the computed square root is the exact square root of a matrix close to a the method is extended for computing the cube root of a matrices exist for which the square root computed by the schur method is ill conditioned but which nonetheless have well conditioned square roots an optimization approach is suggested for computing the well conditioned square roots in these cases

on square roots of m matrices abstract the question of the existence and uniqueness of an m matrix which is a square root of an m matrix is discussed the results are then used to derive some new necessary and sufficient conditions for a real matrix with nonpositive off diagonal elements to be an m matrix

on a homeomorphism between orbit spaces of linear systems and matrix polynomials abstract the orbit space of controllable systems under system similarity and the orbit space of matrix polynomials with determinant degree equal to the order of the state matrix under right equivalence are proved to be homeomorphic when the quotient compact open topology is considered in the latter as a consequence the variation of the finite and left wiener hopf structures under small perturbations of matrix polynomials with fixed degree for their determinants is described

commutators and powers of infinite unitriangular matrices abstract in the paper we consider some commutator type and power type matrix equations in the group ut k of infinite dimensional unitriangular matrices over a field k we introduce a notion of a power outer commutator k m 1 m k x 1 x k and a power engel commutator e k l m 1 m k x y as outer respectively engel commutators modified by allowing powers of letters instead of letters alone for a given infinite unitriangular matrix a we discuss the matrix equations x k a k m 1 m k x 1 x k a and e k l m 1 m k x y a in variables x x 1 x k y as a main result we provide the necessary and sufficient conditions for solvability of these equations

generalized christoffel darboux formula for skew orthogonal polynomials and random matrix theory we obtain a generalized christoffel darboux gcd formula for skew orthogonal polynomials using this we present an alternative derivation of the level density and two point function for gaussian orthogonal ensembles and gaussian symplectic ensembles of random matrices

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

multiple solutions of a critical polyharmonic equation we prove that for n 2m the equation mu u 4m n 2mu on rn has a sequence of nodal finite energy solutions which is unbounded in dm 2 rn this generalizes a classical result of weiyue ding for m 1

a galois approach to mth roots of matrices with rational entries abstract let a be a given n n matrix with rational entries and irreducible characteristic polynomial f x we investigate the galois groups of f x and f x m to find necessary and sufficient conditions for the existence of a solution b to the matrix equation a b m where b is also a matrix with rational entries we do this by finding necessary and sufficient conditions for f x m to have a factor of degree n with rational coefficients

a power method for computing square roots of complex matrices abstract in this paper higher order convergent methods for computing square roots of nonsingular complex matrices are derived these methods are globally convergent and are based on eigenvalue shifting and powering specifically it is shown for each positive integerr 2 a convergent method of orderrcan be developed these algorithms can be used to compute square roots of general nonsingular complex matrices such as computing square roots of matrices with negative eigenvalues

numerical approximation of the product of the square root of a matrix with a vector abstract given an n n symmetric positive definite matrix a and a vector c two numerical methods for approximating a 1 2 c are developed analyzed and computationally tested the first method applies a newton iteration to a specific nonlinear system to approximate a 1 2 c while the second method applies a step control method to numerically solve a specific initial value problem to approximate a 1 2 c assuming that a is first reduced to tridiagonal form the first method requires o n 2 operations per iteration while the second method requires o n operations per iteration in contrast numerical methods that first approximate a 1 2 and then compute a 1 2 c generally require o n 3 operations per iteration

functions of matrices abstract we use the cayley hamilton theorem and the sequence of horner polynomials associated with a polynomial w z to obtain explicit formulas for functions of the form f ta where f is defined by a convergent power series and a is a square matrix we use a well known explicit formula for the resolvent of a and show that f ta is the hadamard product of f t and the function i ta 1 which is easily obtained from the resolvent

extraction of nth roots of 2 2 matrices abstract this paper is concerned with the determination of algebraic formulae giving all the solutions of the matrix equation x n a where n is a positive integer greater than 2 and a is a 2 2 matrix with real or complex elements if a is a 2 2 scalar matrix the equation x n a has infinitely many solutions and we obtain explicit formulae giving all the solutions if a is a non scalar 2 2 matrix the equation x n a has a finite number of solutions and we give a formula expressing all solutions in terms of a and the roots of a suitably defined n th degree polynomial in a single variable this leads to very simple formulae for all the solutions when a is either a singular matrix or a non singular matrix with two coincident eigenvalues similarly when n 3 or 4 we get explicit algebraic formulae for all the solutions we also determine the precise number of solutions in various cases

computation of roots of real and complex matrices an algorithm is presented in this paper by which the rth root of real or complex matrices can be found without the computation of the eigenvalues and eigenvectors of the matrix all required computations are in the real domain the method is based on the newton raphson algorithm and is capable of finding roots even when the matrix is defective computing the root of a matrix from eigenvalues and eigenvectors would be the preferred method if these data were available

the algebraic structure of pencils and block toeplitz matrices abstract we restate several results on the kronecker structure obtainable after a given matrix pencil is affected by an arbitrarily small perturbation and state some new results on the kronecker structure after a new row is appended or a column deleted the results are simple consequences of a theory of majorization for semi infinite integer sequences the bounds after a new row is appended correspond to bounding the controllability indices after adding a single input to a linear time invariant dynamical system we illustrate the known relations between the kronecker indices the segre characteristics and the nullities of certain block toeplitz matrices constructed from the original pencils and discuss how those nullities change when the pencil is affected by a perturbation or a new row

computing primary solutions of equations involving primary matrix functions abstract the matrix equation f x a where f is an analytic function and a is a square matrix is considered some results on the classification of solutions are provided when f is rational a numerical algorithm is proposed to compute all solutions that can be written as a polynomial of a for real data the algorithm yields the real solutions using only real arithmetic numerical experiments show that the algorithm performs in a stable fashion when run in finite precision arithmetic

a riemann hilbert problem for skew orthogonal polynomials we find a local d 1 x d 1 riemann hilbert problem characterizing the skew orthogonal polynomials associated to the partition function of orthogonal ensembles of random matrices with a potential function of degree d our riemann hilbert problem is similar to a local dxd riemann hilbert problem found by kuijlaars and mclaughlin characterizing the bi orthogonal polynomials this gives more motivation for finding methods to compute asymptotics of high order riemann hilbert problems and brings us closer to finding full asymptotic expansions of the skew orthogonal polynomials

structure of the nth roots of a matrix abstract let k be a subfield of c we give a criterion for a nonsingular matrix a in mmk to have an nth root in mmk the number of similarity classes of nth roots of a is given for k r further we indicate which matrices a in mmk have infinitely many nth roots in mmk

nodal solutions for the fractional yamabe problem on heisenberg groups we prove that the fractional yamabe equation rm cal l gamma u vert u vert 4 gamma q 2 gamma u on the heisenberg group n has n 1 2 sequences of nodal sign changing weak solutions whose elements have mutually different nodal properties where rm cal l gamma denotes the cr fractional sub laplacian operator on n q 2n 2 is the homogeneous dimension of n and gamma in bigcup nolimits k 1 n k kq q 1 our argument is variational based on a ding type conformal pulling back transformation of the original problem into a problem on the cr sphere s2n 1 combined with a suitable hebey vaugon type compactness result and group theoretical constructions for special subgroups of the unitary group u n 1

similarity and block similarity we present some equivalent characterizations of the block similarity of matrix pairs with an application to a characterization of the similarity of two square matrices also a survey of some generalizations of properties of the invariants of similarity to the invariants of block similarity is presented

two results about the matrix exponential abstract two results about the matrix exponential are given one characterizes the matrices a which satisfy e a e a h e a h e a another gives upper bounds on trace e a e a h when a is stable the bounds preserve the asymptotic stability