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

on the classification of clifford algebras and their relation to spinors in n dimensions a classification of all the clifford algebras is given in terms of kronecker products of the quaternion and dihedral groups the relationship to spinors in n dimensions is explicitly determined we show that the real clifford algebra in minkowski spacetime is distinct from both the algebra of dirac matrices and the algebra of majorana matrices and cannot be realized by the spinor framework the matrix representations of clifford algebras are discussed and are utilized to give a classification of the real forms of lie algebras we are thus able to relate clifford lie and spinor algebras in an intrinsic geometrical setting

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

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

from godel quaternions to nonlinear sigma models from the ring of godel quaternions the authors construct new quaternionic groups and a pseudo hopf fibration these can be used to study sigma models valued in a four dimensional hyperboloid particular solutions of these models are also given

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

realization extension and classification of certain physically important groups and algebras an associative algebra of differential forms with division has been constructed the algebra of forms in each different space provides a practical realization of the universal clifford algebra of that space a classification of all such algebras is given in terms of two distinct types of algebras nk and sk the former include the dihedral quaternion and majorana algebras the latter include the complex spinor and dirac algebras the associative product expresses hodge duality as multiplication by a basis element this makes possible the realization of higher order algebras in a calculationally useful algebraic setting the fact that the associative algebras as well as the enveloped lie algebras are precisely those arising in physics suggests that this formalism may be a convenient setting for the formulation of basic physical laws

an algebraic and geometric approach to non bijective quadratic transformations this work is concerned with an extension of the so called levi civita and kustaanheimo stiefel transformations the extension is achieved along two lines firstly the authors examine the latter transformations in other dimensions than the ones originally considered by on the one hand levi civita and on the other hand by kustaanheimo and stiefel secondly they pass from the compact to the non compact case this leads to quadratic non bijective transformations that they refer to as hurwitz or kustaanheimo stiefel like and quasihurwitz or levi civita like transformations the hurwitz and quasihurwitz transformations are introduced and studied in an algebraic framework which relies on the use of eight dimensional cayley dickson algebras an explicit formulation of the hurwitz and quasihurwitz transformations is also given in terms of clifford algebras the hurwitz transformations are investigated from a geometrical viewpoint indeed they are connected to hopf and pseudohopf fibrations finally some differential aspects of the hurwitz and to a lesser extent quasihurwitz transformations are developed in view of future physical applications

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

electromagnetism and the holomorphic properties of spacetime the cauchy riemann equations of holomorphy are extended to fields in higher dimensional spaces in a framework of clifford algebras the equations of holomorphy in minkowski spacetime turn out to be the maxwell equations in vacuum the lorentz gauge condition is a result of the holomorphy sources can be included in an extension of the residue theorem where charges correspond to the residues

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

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

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

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 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

the exponential mapping in clifford algebras the exponential mapping takes the lie algebra of the lorentz group into the lorentz group each element of the group is defined as a formal power series while the product of two exponential elements usually involves an infinite sum of commutator terms such as in the baker campbell hausdorff formula because of the special arithmetic in clifford algebras many baker campbell hausdorff like formulas and identities can be calculated or summed exactly when they involve only elements from the algebra we calculate exact identities for the baker campbell hausdorff formula and related formulas in the quaternion and dihedral algebras these are useful in treatments of the lorentz group and make possible a truly finite as opposed to infinitesimal description of a transformation group in physics

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

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

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

on the use of differentials in statistics abstract this paper explores the connections between symbolic matrix derivatives as developed in the statistical literature and the mathematical theory of differentials it is shown how many published results on symbolic matrix derivatives can be obtained by straightforward algebraic manipulations monotone operators and quadratic differential forms are used to solve certain extreme value problems in statistics in a simple and rigorous way

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

a bbgky hierarchy for the extended kinetic theory abstract the extended boltzmann equation which considers the possibility of creation and annihilation of particles by collision events is derived from a generalized liouville equation through the construction of the corresponding bbgky hierarchy this derivation requires a generalization of the concept of probability density and reduced distribution functions for the case in which the number of particles is not preserved this extended boltzmann equation is compared with previously proposed models

a generalization of the construction of ilamed and salingaros ilamed and salingaros construct the real and complex algebras with three anticommuting elements which can arise in physics it is shown here that the algebra of color can be similarly constructed with six anticommuting elements as a consequence of this construction these algebras are all simple quadratic algebras

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

algebras with three anticommuting elements i spinors and quaternions a general construction of alternative algebras with three anticommuting elements and a unit is given as an exhaustive result over the real and complex fields we obtain the clifford algebras h quaternions n1 dihedral clifford algebra which is related to real 2 spinors and s1 algebra of pauli matrices which is related to complex 2 spinors what is important is that the algebras n1 and s1 possess inverses everywhere except on a region akin to the light cone of the minkowski space while the quaternion algebra h has inverses everywhere except at the zero element we discuss the reasons why the three algebras n1 h and s1 are so difficult to distinguish in the representation space of 2 2 complex matrices

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

a construction relating clifford algebras and cayley dickson algebras a review of the applications of the octonions in physics is given a construction is presented both the cayley dickson algebras and the clifford algebras arise naturally under this construction from the quaternion algebras the mathematical properties of the algebras constructed are discussed

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

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

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

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 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

quadratic alternative algebras quadratic alternative algebras are completely classified up to isomorphism by means of the associated set of vectors they include the quaternion and octonion algebras but also many other nonassociative algebras used in physics

reducibility of a class of nonlinear kinetic integral equations the reducibility to differential forms of certain nonlinear integral operators appearing in kinetic theory is discussed in the frame of a stationary problem in gas dynamics with removal events a wide class of transition kernels allowing such reducibility is characterized and the validity of these kernels as approximations to real ones is studied in terms of their asymptotic behaviour for large and small velocities in fact this behaviour is shown to determine the corresponding limiting velocity dependence in the solutions to the kinetic equation

algebras with anticommuting basal elements space time symmetries and quantum theory a collection a of algebras with anticommuting basal elements is investigated it is shown that the collection a includes the quaternions the octonions the algebra of color as well as other algebras familiar to the physicist each algebra a is a quadratic jordan admissible algebra and possesses a norm that is a generalization of the minkowski metric using a cayley dickson like process each such algebra a in a can be embedded into a larger algebra a that is also in the collection a these algebras should provide candidates for models to describe observables color and other phenomena encountered in particle physics

a hierarchical kinetic theory of birth death and fission in age structured interacting populations we develop mathematical models describing the evolution of stochastic age structured populations after reviewing existing approaches we formulate a complete kinetic framework for age structured interacting populations undergoing birth death and fission processes in spatially dependent environments we define the full probability density for the population size age chart and find results under specific conditions connections with more classical models are also explicitly derived in particular we show that factorial moments for non interacting processes are described by a natural generalization of the mckendrick von foerster equation which describes mean field deterministic behavior our approach utilizes mixed type multidimensional probability distributions similar to those employed in the study of gas kinetics and with terms that satisfy bbgky like equation hierarchies

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

algebras with three anticommuting elements ii two algebras over a singular field the clifford algebra generated by the elements 1 with 2 1 is an abelian ring of dimension two with properties analogous to the complex field c the ring has a string of singular inverses and may be regarded as a singular field which circumvents both the fundamental theorem of algebra and the frobenius theorem we construct two associative algebras of dimension four over the clifford algebra 1 and the biquaternions of clifford 2 and demonstrate that both algebras possess inverses everywhere except on a singular region akin to the light cone of the minkowski space matrix representations are discussed as well as the importance of the algebras 1 and 2 in the description of physical vector fields

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

kinetic theory of age structured stochastic birth death processes classical age structured mass action models such as the mckendrick von foerster equation have been extensively studied but are unable to describe stochastic fluctuations or population size dependent birth and death rates stochastic theories that treat semi markov age dependent processes using e g the bellman harris equation do not resolve a population s age structure and are unable to quantify population size dependencies conversely current theories that include size dependent population dynamics e g mathematical models that include carrying capacity such as the logistic equation cannot be easily extended to take into account age dependent birth and death rates in this paper we present a systematic derivation of a new fully stochastic kinetic theory for interacting age structured populations by defining multiparticle probability density functions we derive a hierarchy of kinetic equations for the stochastic evolution of an aging population undergoing birth and death we show that the fully stochastic age dependent birth death process precludes factorization of the corresponding probability densities which then must be solved by using a bogoliubov born green kirkwood yvon like hierarchy explicit solutions are derived in three limits no birth no death and steady state these are then compared with their corresponding mean field results our results generalize both deterministic models and existing master equation approaches by providing an intuitive and efficient way to simultaneously model age and population dependent stochastic dynamics applicable to the study of demography stem cell dynamics and disease evolution

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

the metric of a space and quadratic algebras a collection of algebras with product that preserves the metric of the underlying vector space is investigated it is shown that this collection includes the generalized quaternions the composition algebras and the algebra of color all the algebras in this collection that are either commutative or alternative are determined the ramifications of nondegenerate metrics are detailed if the algebra is alternative and not of dimension 1 2 4 or 8 over the base field the metric must be degenerate

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

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