products of projections and positive operators this article is devoted to the study of the set t of all products pa with p an orthogonal projection and a a positive semidefinite operator we describe this set and study optimal factorizations we also relate this factorization with the notion of compatibility and explore the polar decomposition of the operators in t

decomposition of matrices into three involutions abstract in this paper we generalize some results of c s ballantine concerning products of three n n complex involutions we prove that each n n complex matrix a with determinant 1 and dimker a n 2 for all c is the product of three involutions on the other hand we show that if an n n complex matrix a is the product of three involutions then m 2n r 3 and m 3n 4 where m dimker a and r dimker a 3 for any 0 and 4 1 we also completely characterize products of three 5 5 complex involutions

alternating forms and self adjoint operators abstract let k be a field and n 1 an integer we study the action of the symplectic group over k on the set of alternating forms on k 2 n we show that the action on pairs of forms can be interpreted in terms of the conjugation action on self adjoint operators and obtain some old and new results using this interpretation in particular for each k and n we determine the smallest base size for the action of pgl 2 n k on the set of non degenerate skew symmetric matrices over the field k modulo scalars

when does a linear map belong to at least one orthogonal or symplectic group abstract given an endomorphism u of a finite dimensional vector space over an arbitrary field k we give necessary and sufficient conditions for the existence of a regular quadratic form respectively a symplectic form for which u is orthogonal respectively symplectic since a solution to this problem is already known in the case char k 2 our main contribution lies in the case char k 2 when char k 2 we also give necessary and sufficient conditions for the existence of a regular symmetric bilinear form for which u is orthogonal when k is finite with characteristic 2 we give necessary and sufficient conditions for the existence of an hyperbolic quadratic form respectively a regular non hyperbolic quadratic form respectively a regular nonalternate symmetric bilinear form for which u is orthogonal

decomposition of matrices into 2 involutions abstract we consider the group slnf of all n by n matrices with determinant 1 over a field let res a denote rank a i for any a in slnf a matrix a is called a k involution if a2 i and res a k we prove that every matrix a in slnf is the product of at most res a 1 2 2 2 involutions for n 3 if char f 2 and for n 4 if char f 2

a mathematical model of adomian polynomials the adomian decomposition method is reviewed and a mathematical model of adomian polynomials is introduced to show the convenience of this model for computer programming the mathematica package is used to generate the adomian polynomials for any order

factorizations of integer matrices as products of idempotents and nilpotents abstract it is proved that for n 3 every n n matrix with integer entries and determinant zero is the product of 36n 217 idempotent matrices with integer entries

baer levi semigroups of linear transformations given an infinite dimensional vector space v we consider the semigroup gs m n consisting of all injective linear v v for which codim ran n where dimv m n 0 this is a linear version of the well known baer levi semigroup bl p q defined on an infinite set x where x p q 0 we show that although the basic properties of gs m n are the same as those of bl p q the two semigroups are never isomorphic we also determine all left ideals of gs m n and some of its maximal subsemigroups in this we follow previous work on bl p q by sutov 1966 and sullivan 1978 as well as levi and wood 1984

products of square zero operators we characterize matrices that can be written as a product of two or three square zero matrices we also consider the same questions for bounded operators on an infinite dimensional separable complex hilbert space and in the calkin algebra

an application of the adomian decomposition method to the transient behavior of a model biochemical reaction abstract an approximate analytical solution for the transient phase of the michaelis menten reaction is derived using the adomian decomposition method the analytical solution which is given in the form of a power series is found to be highly accurate in predicting the behavior of the reaction in the very early stages to accelerate the convergence of the power series solution and extend its region of applicability throughout the entire transient phase we have used a the method of pade approximants and b the iterated shanks transformation both the pade approximant and the shanks transformation are shown to converge rapidly throughout and beyond the transient period and yield very accurate results a comparison of the various analytical approximations and a direct numerical solution of the nonlinear initial value problem is also presented

product of two positive contractions abstract several characterizations are given for a square matrix that can be written as the product of two positive semidefinite contractions based on one of these characterizations and the theory of alternating projections a matlab program is written to check the condition and construct the two positive contractions whose product equal to the given matrix if they exist

sums and products of quasi nilpotent operators it is proved that a bounded operator on hilbert space is the sum of two quasi nilpotent operators if and only if it is not a non zero scalar plus a compact operator necessary conditions and sufficient conditions for an operator to be the product of two quasi nilpotent operators are given

expressing infinite matrices as products of involutions abstract we take into consideration ut k the group of upper triangular infinite matrices whose entries lying on the main diagonal are equal to either 1 or 1 we show that for any field k every element of this group can be expressed as a product of at most five involutions moreover we prove that if k is a field of characteristic different from 2 then four involutions suffice

nilpotent local automorphisms of an independence algebra let a be a strong independence algebra of infinite rank m let a be the inverse monoid of all local automorphisms of a the baer levi semigroup b over the algebra a is defined to be the subsemigroup of a consisting of all the elements with dom a and corank im m let k m be the subsemigroup of a generated by b 1 b then k m is inverse and it is generated as a semigroup by n 2 the subset of a consisting of all nilpotent elements of index 2 the 2 nilpotent depth 2 k m of k m is defined to be the smallest positive integer t such that k m n 2 n 2 t in fact 2 k m is either 2 or 3 and a criterion is found which distinguishes between the two cases if n denotes the set of all nilpotents in a then the subsemigroup generated by n is also k m in fact k m is proved to be exactly n 2

products of positive semi definite matrices abstract it is known that every complex square matrix with nonnegative determinant is the product of positive semi definite matrices there are characterizations of matrices that require two or five positive semi definite matrices in the product however the characterizations of matrices that require three or four positive semi definite matrices in the product are lacking in this paper we give a complete characterization of these two types of matrices with these results we give an algorithm to determine whether a square matrix can be expressed as the product of k positive semi definite matrices but not fewer for k 1 2 3 4 5

on idempotent affine mappings let v be a vector space and end v the semigroup of endomorphisms of v an affine mapping of v is a map a v v given by x a x a where a belongs to end v and a is some element of v let v be the semigroup of affine mappings of v let e denote the non injective idempotents of end v and let denote the idempotents of v in this paper is determined in terms of e when end v consists of all endomorphisms of v and when end v only contains the continuous endomorphisms in which case we restrict v to being an inner product space

powers of 1 cyclic conjugacy classes in the groups gln f and sln f abstract let a gl n f where f is a field we say that a is 1 cyclic if a is similar to a matrix of the form a diag a 1 a 2 a k where a i gl l i f is cyclic for 1 i k l 1 0 1 and l i 2 for 2 i k it is shown that if a gl n f is 1 cyclic where n 2 and f 4 then every nonscalar matrix m gl n f whose determinant equals det a 4 is the product of four matrices which are similar to a under matrices of sl n f the problem of expressing a scalar matrix as a product of similar 1 cyclic matrices is also discussed the above result is applied to problems of factorizing matrices in the group sl n f into products of unipotent matrices of index 2 and into products of matrices of fixed finite order

commutators and companion matrices over rings of stable rank 1 abstract we consider the group gl n a of all invertible n by n matrices over a ring a satisfying the first bass stable range condition we prove that every matrix is similar to the product of a lower and upper triangular matrix and that it is also the product of two matrices each similar to a companion matrix we use this to show that when n 3 and a is commutative every matrix in sl n a is the product of two commutators

products of involutory matrices over rings abstract let gln a be a matrix over a commutative ring a with 1 such that det 2 1 if is cyclic it can be written as a product of at most three involutions when a satisfies the first bass stable range condition then can be written as a product of at most five involutions if in addition either n 3 or n 4 and det 1 then can be written as a product of at most four involutions when a is a dedekind ring of arithmetic type the number of involutions needed to express is uniformly bounded for any n 3 when a c x the number of involutions is unbounded for any n 2

on the adomian decomposition method and comparisons with picard s method abstract this paper shows that the adomian decomposition method stochastic systems academic press new york 1983 is not to be identified with any of the earlier methods since it solves a wide class of nonlinear and stochastic equations without the often unphysical assumptions which have become customary similarities between the picard method and the decomposition method are purely superficial

factorization of matrices into quadratic ones iii abstract it is shown that every complex n n matrix t is the product of four quadratic matrices moreover if t is invertible then the number of required quadratic matrices can be reduced to three and three is sharp for n 4 complex matrices which are expressible as the product of an idempotent and an invertible quadratic matrix are also characterized

every 2n by 2n complex matrix is a sum of three symplectic matrices abstract let j 2 n 0 i n i n 0 an a m 2 n c is called symplectic if a t j 2 n a j 2 n if n 1 then we show that every matrix in m 2 n c is a sum of two symplectic matrices if n 1 then we show that every matrix in m 2 n c is a sum of three symplectic matrices moreover we show that some matrices cannot be written with less than three symplectic matrices we also show that for every a m 2 n c there exist symplectic p q m 2 n c and b c d m n c such that p a q b c 0 d if a is skew hamiltonian j 2 n 1 a t j 2 n a then we show that a is a sum of two symplectic matrices

products of nilpotent matrices we show that any complex singular square matrix t is a product of two nilpotent matrices a and b with rank a rank b rank t except when t is a 2 x 2 nilpotent matrix of rank one an n x n complex matrix t is nilpotent if t 0 it is easily seen that a product of finitely many nilpotent matrices must be singular the purpose of this note is to prove the converse

products of matrices with prescribed spectra and ranks this paper studies the possibility of writing a given square matrix as the product of two matrices with prescribed spectra and ranks it extends some previously known results

on some factorizations of operators abstract given two subsets a and b of the algebra of bounded linear operators on a hilbert space h we denote by a b ab a a b b the goal of this article is to describe a b if a and b denote classes of projections partial isometries positive semidefinite operators etc moreover fixed t a b we shall describe a b t a b a b ab t p 1 a b t a a t ab for some b b and p 2 a b t b b t ab for some a a

the operator factorization problems we survey various results concerning operator factorization problems more precisely we consider the following setting let h be a complex hilbert space and let g h be the algebra of all bounded linear operators on h for a given subset of 3 h we are interested in the characterization of operators in q h which are expressible as a product of finitely many operators in v and for each such operator the minimal number of factors in a factorization the classes of operators we consider include normal operators involutions partial isometries together with their various subclasses and other miscellaneous classes of operators most of the known results are for operators on finite dimensional spaces or finite matrices the paper concludes with some applications due to hochwald concerning the uniqueness of the adjoint operation on operators

products of idempotent linear transformations in 1966 j m howie characterised the transformations of an arbitrary set that can be written as a product under composition of idempotent transformations of the same set in 1967 j a erdos considered the analogous problem for linear transformations of a finite dimensional vector space and in 1983 r j dawlings investigated the corresponding idea for bounded operators on a separable hilbert space in this paper we study the case of arbitrary vector spaces

the length problem for a sum of idempotents abstract it was shown by r e hartwig and m s putcha and independently by p y wu that the complex square matrix t is a sum of finitely many idempotent matrices if and only if tr t is an integer and tr t rank t we determine the minimal number of required idempotents for such matrices in terms of their traces and sizes and we completely solve the problem for matrices t with size 2 3 4 or 5 also we give some sufficient necessary conditions for a complex matrix t to be the sum of three or more idempotents

report on the dublin matrix theory conference march 1984 matrices over finite fields graciano de oliveira departamento de matematica universidad de coimbra coimbra portugal

products of cyclic similarity classes in the groups gln f abstract let g gln f be the group of all n n invertible matrices over a field f where f 4 and n 3 let j1 j2 be two prescribed cyclic elements of g in jordan canonical form and assume that all the eigenvalues of j1 or j2 lie in f it is shown that any nonscalar matrix in g can be written as a product of two matrices having the prescribed jordan canonical forms j1 j2 subject only to the obvious 0 determinant condition as a corollary we show that if f 4 the group psln f has a conjugacy class c such that c2 psln f another corollary states that every nonscalar matrix in g can be written as a product of two cyclic unipotent matrices

linear maps preserving products of positive or hermitian matrices abstract in this note we consider linear maps on the complex n n matrix algebra m n preserving products of either two positive matrices a positive matrix and a hermitian matrix or two hermitian matrices we give characterizations of all those injections

products of projections in von neumann algebras we describe the elements of von neumann algebras which can be represented as products of orthogonal projections and idempotents and estimate the minimal number of terms in the product

multiplicative maps on matrices that preserve the spectrum abstract let m n denote the set of all n n matrices over the complex numbers n 2 let a n m n be either the set of all invertible matrices the set of all unitary matrices or a multiplicative semigroup containing the singular matrices theorem if a n m n is a spectrum preserving multiplicative map then there exists a matrix r in m n such that s r 1 sr for all s in a n

a computational method for solving singularly perturbed turning point problems exhibiting twin boundary layers singularly perturbed turning point problems tpps for second order ordinary differential equations des exhibiting twin boundary layers are considered in order to obtain numerical solution of these problems a computational method is suggested in which exponentially fitted difference schemes are combined with classical numerical methods in this method the given internal domain of definition of differential equation is divided into four subintervals and we solve the differential equation in each interval and combine the solution the implementation of the method on parallel architectures is mentioned error estimates are derived and numerical examples are given

on the number of invariant factors of matrix products we prove an inequality relating the number of nontrivial invariant factors of n n matrices a and b with those of ab and get some results on the cases of equality in particular we characterize the similarity classes a and b with all eigenvalues in the base field such that ab is nilpotent for some a a and b b

factorizations of upper triangular matrices if d is an integral domain in which every element can be factored as a product of irreducible elements then every element a tn d the semigroup of upper triangular matrices with nonzero determinant can be factored as a product of irreducible elements we classify both the irreducible elements and units of this cancellative semigroup having achieved this we provide a means of measuring the uniqueness or non uniqueness of these factorizations in terms of the uniqueness or non uniqueness of factorizations in d in addition to using well studied invariants from factorization theory of commutative semigroups we introduce a new tool weak transfer homomorphisms which are more appropriate for noncommutative settings

semigroups generated by nilpotent transformations in a seminal paper published in 1966 john howie characterised the elements of tx the semigroup under composition of all total transformations of a set x into itself which can be written as a product of idempotents in tx we now initiate the study of the subsemigroup of px the semigroup of all partial transformations of x which is generated by the nilpotents of px

products of cyclic conjugacy classes in the groups psl n f abstract let g be the group psl n f where f is a field and n 3 if c is a conjugacy class of g denote c 1 c 1 c c the following is proved 1 if c1 c2 c3 are cyclic conjugacy classes of g then c1c2c3 g 1g 2 if f is algebraically closed and c1 c2 are cyclic conjugacy classes of g then c1c2 g if and only if c1 c 12 generalizations of these results concerning factorizations of a given nonscalar invertible matrix as a product of two or three cyclic matrices each lying in a prescribed conjugacy class of gl n f or sl n f are discussed

factorizations of operator matrices abstract this note gives explicit factorizations of a 2 2 operator matrix as a product of an upper triangular operator matrix and an involutory unitary or j unitary operator matrix a pattern is given for construction of factorizations of this kind

each symplectic matrix is a product of four symplectic involutions abstract gustafson halmos and radjavi in 1973 proved that each matrix a with det a 1 is a product of four involutions we prove that these involutions can be taken to be symplectic if a is symplectic every symplectic matrix has unit determinant using this result we give an alternative proof of laffey s theorem that every nonsingular even size matrix is a product of skew symmetric matrices ballantine in 1978 proved that each matrix a with det a 1 is a product of four coninvolutions we prove that these coninvolutions can be taken to be symplectic if a is symplectic we also prove that each hamiltonian matrix is a sum of two square zero hamiltonian matrices

expressing infinite matrices over rings as products of involutions abstract let k be an arbitrary field and r be an arbitrary associative ring with identity 1 slowik in 12 proved that each matrix of ut k the group of upper triangular infinite matrices whose entries lying on the main diagonal are equal to either 1 or 1 can be expressed as a product of at most five involutions in this article we extend the investigate to an arbitrary associative ring r with identity 1 our conclusion is that every element of ut r can be expressed as a product of at most four involutions we also prove that for the complex field every element of t c the group of upper triangular infinite matrices whose entries lying on the main diagonal satisfy a a 1 can be expressed as a product of at most three coninvolutions matrices satisfying a a i

some open problems in matrix theory arising in linear systems and control control theory has long provided a rich source of motivation for developments in matrix theory accordingly we discuss some open problems in matrix theory arising from theoretical and practical issues in linear systems theory and feedback control the problems discussed include robust stability matrix exponent induced norms stabilizability and pole assignability and nonstandard matrix equations a substantial number of references are included to acquaint matrix theorists with problems and trends in this application area feedback control theory has long provided a rich source of motivation for developments in matrix theory the purpose of this paper is to discuss several open problems in matrix theory that arise from theoretical and practical issues in feedback control theory and the associated area of linear systems theory many of these problems are remarkably simple to state are of intense interest in control theory and applications and yet remain unsolved besides leading to the resolution of these problems it is hoped that this paper will help to stimulate increased interaction between matrix and control theorists accord

every real symplectic matrix is a product of real symplectic involutions abstract a 2n by 2n matrix a is symplectic if a t 0 i i 0 a 0 i i 0 it is known that if n 1 then every 2n by 2n complex symplectic matrix is a product of four symplectic involutions we consider the real case we give an example of a real symplectic matrix which is a product of two complex symplectic involutions but is not a product of two real symplectic involutions we show that every 4 by 4 real symplectic matrix is a product of four real symplectic involutions we use this result to show that for n 1 every 2n by 2n real symplectic matrix is a product of a finite number of real symplectic involutions

products of diagonalizable matrices over a field of characteristic two abstract it is shown that every square matrix over a characteristic two field with at least four elements is a product of two diagonalizable matrices

decomposition of a scalar operator into a product of unitary operators with two points in spectrum abstract we consider products of unitary operators with at most two points in their spectra 1 and e i we prove that the scalar operator e i i is a product of k such operators if 1 1 k 3 k 1 1 k 3 for k 5 also we prove that for e i 1 only a countable number of scalar operators can be decomposed in a product of four operators from the mentioned class as a corollary we show that every unitary operator on an infinite dimensional space is a product of finitely many such operators

the product of matrix subspaces abstract in factoring matrices into the product of two matrices operations are typically performed with elements restricted to matrix subspaces such modest structural assumptions are realistic for example in large scale computations this paper is concerned with analyzing associated matrix geometries curvature of the product of two matrix subspaces is assessed the case of vanishing curvature gives rise to the notion of factorizable matrix subspace this can be regarded as an analogue of the internal zappa szep product in group theory interpreted in this way several classical instances are encompassed by this structure the craig sakamoto theorem fits naturally into this framework

products of idempotents in regular rings ii abstract this paper characterizes products of idempotents in von neumann regular rings which are unit regular or right self injective for unit regular rings the minimum number of idempotents needed in such a product is determined thereby generalizing a result of c s ballantine products of idempotent matrices linear algebra appl 19 1978 81 86 in the case of a matrix with entries from a field

symmetric bilinear forms and vertices in characteristic 2 let g be a finite group and let k be an algebraically closed field of characteristic 2 suppose that m is an indecomposable kg module which affords a non degenerate g invariant symmetric bilinear form we assign to m a collection of 2 subgroups of g called its symmetric vertices each of which contains a green vertex of m with index at most 2 if m is irreducible then its symmetric vertices are uniquely determined up to g conjugacy if b is the real 2 block of g containing m we show that each symmetric vertex of m is contained in an extended defect group of b moreover we characterise the extended defect groups in terms of symmetric vertices in order to prove these results we develop the theory of involutary g algebras this allows us to translate questions about symmetric kg modules into questions about projective modules of quadratic type

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

adomian s method of decomposition critical review and examples of failure abstract adomian s method of decomposition is considered in application to initial boundary value problems for the one space dimensional spatially homogeneous heat conduction equation it is shown that the fundamental equation of the method is well defined only for certain restricted types of boundary conditions within the class of such boundary conditions examples are given such that the fundamental equation fails to have a unique solution and such that the sequence produced by iteration of this equation is divergent the latter is a counterexample to a published assertion of convergence

multigrid applied to singular perturbation problems the solution of the singular perturbation problem email protected u b x u f 0 0 u 0 u 0 u 1 u 1 by a multigrid algorithm is considered theoretical and experimental results for a number of different discretizations are presented the theoretical and observed rates agree with the results developed in an earlier work of kamowitz and parter in addition the rate of convergence of the algorithm when the coarse grid operator is the natural finite difference analogue of the fine grid operator is presented this is in contrast to the case in the previous work where the galerkin choice i h hl hi h h was used for the coarse grid operators

pairs of alternating forms and products of two skew symmetric matrices let k be a field and let s t s1 t1 be skew symmetric matrices over k with s s1 both nonsingular if k has characteristic 2 a skew symmetric matrix is a symmetric one with zero diagonal it is shown that there exists a nonsingular matrix p over k with p sp s1 p tp t1 where p denotes the transpose of p if and only if s 1t and s 11t1 are similar it is also shown that a 2m 2m matrix over k can be factored as st with s t skew symmetric and s nonsingular if and only if a is similar to a matrix direct sum b b where b is an m m matrix over k this is equivalent to saying that all elementary divisors of a occur with even multiplicity an extension of this result giving necessary and sufficient conditions for a square matrix to be so expressible without assuming that either s or t is nonsingular is included

products of involutions in the stable general linear group abstract in the stable general linear group over an arbitrary field we prove that every element with determinant 1 is the product of three involutions and of no less in general we also obtain several results of the same flavor with applications to decompositions of automorphisms of an infinite dimensional vector space that are scalar multiples of finite rank perturbations of the identity

products of symplectic normal matrices abstract a matrix a m 2 n c is symplectic if a t 0 i n i n 0 a 0 i n i n 0 we show that every symplectic matrix is a product of a symplectic unitary and a symplectic skew hermitian matrix we show that every symplectic matrix is a product of four symplectic skew hermitian matrices or a product of four symplectic hermitian matrices we give the possible jordan canonical forms of symplectic matrices which can be written as a product of a symplectic hermitian and a matrix which is either symplectic hermitian or symplectic skew hermitian

characterizations of products of symmetric matrices abstract characterizations are obtained for matrices c of the form c a where a are n n matrices over the real field such that a is symmetric and c is nonnegative definite among others a proof of recent generalization of cochran s theorem is given

systems of nonlinear partial differential equations abstract the author s decomposition method 1 3 is applied to a system of nonlinear partial differential equations to demonstrate potential applicability to such systems

multigrid methods for variational problems this paper develops a very simple but powerful theory for multigrid methods which applies directly to variationally posed operator equations

products of positive semidefinite matrices abstract we characterize the complex square matrices which are expressible as the product of finitely many positive semidefinite matrices a matrix t can be expressed as such if and only if det t 0 moreover the number of factors can always be limited to five we also determine those matrices which can be expressed as the product of two or four positive semidefinite matrices these results are analogous to the ones obtained before by c s ballantine for products of positive definite matrices

products of unipotent matrices of index 2 abstract we show that an n n complex matrix t is the product of two unipotent matrices of index 2 if and only if t is similar to a matrix of the form d d 1 i n i mi 1 ji where 0 and 1 are not eigenvalues of d n is nilpotent and each ji is a nilpotent jordan block of even size on the other hand t is the product of finitely many unipotent matrices of index 2 if and only if det t 1 in this case the minimal number of required unipotents is 1 if n 1 3 if n 2 and 4 if n 3

a study of some multigrid ideas in an effort to understand certain ideas and concepts associated with multigrid iterations we give an in depth study of a particular simple problem we consider a standard finite difference system associated with the two point boundary value problem pu bu qu 0 u 0 u 1 0 the operators i 2 h h i h 2 h are operator based interpolation and projection operators while the smoothers are the damped jacobi iterations with parameter a 0 we determine the exact rates of convergence for the two grid scheme and upper bounds 1 for the multigrid schemes experimental results are discussed

products of diagonalizable matrices abstract let f denote a field such that char f 2 it is shown that every square matrix over f is expressible as a product of two diagonalizable matrices except when f gf 3 in which case three diagonalizable matrices are needed in general partial results for the case where char f 2 is also presented finally the extent to which the nullities of these factors can be prescribed is also investigated

the jordan cononical form of a product of a hermitian and a positive semidefinite matrix abstract a given square complex matrix c is the product of a positive semidefinite matrix a and a hermitian matrix b if and only if c 2 is diagonalizable and has nonnegative eigenvalues this condition is equivalent to requiring that c have real eigenvalues and a jordan canonical form that is diagonal except for r copies of a 2 by 2 nilpotent jordan block we show that r is bounded from above by the rank of a the nullity of a and both the positive and negative inertia of b it follows that a product of two positive semidefinite matrices is diagonalizable and has nonnegative eigenvalues a result that leads to a characterization of the possible concanonical forms of a positive semidefinite matrix

note on the factorization of a square matrix into two hermitian or symmetric matrices this paper presents elementary proofs of the factorization of a square matrix into two hermitian or symmetric matrices

factorizations of a matrix into symmetric matrices with prescribed nullities abstract necessary and sufficient conditions are presented for a square matrix over an arbitrary field to be a product of k 2 symmetric matrices with prescribed nullities a generalization of the case k 2 to hermitian matrices is also presented

decomposition of infinite matrices into products of commutators of involutions abstract let r be an associative ring with identity 1 in which 2 is invertible we prove that each matrix of ut r the group of upper triangular infinite matrices whose entries lying on the main diagonal are equal to 1 can be expressed as a product of at most two commutators of involutions let f be an arbitrary field with char f 2 using this result each element of the commutator subgroup of the vershik kerov group over f can be decomposed into a product of at most two commutators of involutions