nullity invariance for pivot and the interlace polynomial we show that the effect of principal pivot transform on the nullity values of the principal submatrices of a given square matrix is described by the symmetric difference operator for sets we consider its consequences for graphs and in particular generalize the recursive relation of the interlace polynomial and simplify its proof

orthogonal polynomials and the finite toda lattice the choice of a finitely supported distribution is viewed as a degenerate bilinear form on the polynomials in the spectral parameter z and the matrix representing multiplication by z in terms of an orthogonal basis is constructed it is then shown that the same induced time dependence for finitely supported distributions which gives the ith kp flow under the dual isomorphism induces the ith flow of the toda hierarchy on the matrix the corresponding solution is an n particle finite nonperiodic toda solution where n is the cardinality of the support of c plus the sum of the orders of the highest derivative taken at each point

the numerical range of products of normal matrices abstract in an earlier paper the author developed a formula for the trace class multiplier norm of a matrix of rank at most 2 in this article applications of this formula are given in the main result we suppose that f 1 f n and g 1 g n are given sets of complex numbers a description is given of the union of the numerical ranges of the product fg as f and g run over all n n normal matrices with the given sets as eigenvalues

unicity of minimal rank completions for tri diagonal partial block matrices abstract in this paper we establish necessary and sufficient conditions for a minimal rank completion of a tri diagonal partial block matrix to be unique the case of 4 4 block matrices with four consecutive diagonals is also considered along with its application to the problem of completing a matrix and its inverse

the block independence in the generalized inverse at s 2 for some ordered matrices and applications abstract in this paper the definition of block independence in the generalized inverse a t s 2 is firstly given and then a necessary and sufficient condition for some ordered matrices to be block independent in the generalized inverse a t s 2 is derived as an application a necessary and sufficient condition for a 1 a 2 a k t s 2 a 1 t 1 s 1 2 a 2 t 2 s 2 2 a k t k s k 2 is proved moreover some results are shown with respect to the moore penrose inverse the weighted moore penrose inverse and the drazin inverse respectively

geometry structures of the generalized inverses of block two by two matrices abstract in this paper we study the invertibility and the moore penrose invertibility of 2 2 block matrix with schur complement having closed range the necessary and sufficient conditions for the existence as well as the expressions for the inverses and the moore penrose inverses are obtained

rank preserving diagonal completions of a matrix abstract suppose a is an n by n matrix over a field f we prove that it is possible to complete the diagonal entries of a so that the resulting rank of a is as small as possible when n 3r where r is the off diagonal rank of a and n r 3 1

structure ranks of matrices abstract the structure rank of a matrix i e the maximum order of a nonsingular submatrix all of whose entries are located in a given structure a subset of m n where m n are the row and column index sets respectively were previously studied for the off diagonal part here results on more general ranks e g off block diagonal strictly upper triangular and strictly upper block triangular are presented in particular it is shown that for a nonsingular matrix a in each of the last mentioned three cases the ranks of a and a 1 coincide

outer inverses jacobi type identities and nullities of submatrices abstract according to the jacobi identity if a is an invertible matrix then any minor of a 1 equals up to a sign the determinant of a 1 times the complementary minor in the transpose of a the identity is extended to any outer inverse thereby generalizing several results in the literature for special generalized inverses a permanental analog of the jacobi identity is proved bounds are obtained for the difference between the nullity of a submatrix of a and that of the complementary submatrix in any generalized inverse or an outer inverse of a the result extends earlier work of fiedler markham and gustafson for the inverse and of robinson for the moore penrose inverse

on the sutherland spin model of b n type and its associated spin chain the b n hyperbolic sutherland spin model is expressed in terms of a suitable set of commuting dunkl operators this fact is exploited to derive a complete family of commuting integrals of motion of the model thus establishing its integrability the dunkl operators are shown to possess a common flag of invariant finite dimensional linear spaces of smooth scalar functions this implies that the hamiltonian of the model preserves a corresponding flag of smooth spin functions the discrete spectrum of the restriction of the hamiltonian to this spin flag is explicitly computed by triangularization the integrability of the hyperbolic sutherland spin chain of b n type associated with the dynamical model is proved using polychronakos s freezing trick

structures preserved by generalized inversion and schur complementation in this paper we investigate the inheritance of certain structures under generalized matrix inversion these structures contain the case of rank structures and the case of displacement structures we do this in an intertwined way in the sense that we develop an argument that can be used for deriving the results for displacement structures from thoses for rank structures we pay particular attention to the moore penrose generalized inverse showing that for the cases of most interest the ranks of the structure satisfied by the moore penrose inverse can at most double with respect to the original ranks we consider also the case of inheritance of structure by generalized schur complements

characterizations of the commutators and the anticommutator involving idempotents in this paper we characterize the idempotency the invertibility or generalized invertibility the positivity and the range relations of the commutator c pq qp and the anticommutator d pq qp involving idempotents particular attention is paid to the idempotent operators p and q and some further characteristics when p and q are positive are derived as well

nullities of submatrices of the moore penrose inverse let a be a complex m n matrix of rank r and with moore penrose inverse a if a a11a21a12a22 a b11b21b12b22 are partitioned so that aii is mi ni and bii is ni mi with m m1 m2 n n1 n 2 and if the row nullity of a matrix is denoted by then m r b22 a11 n r

unique minimal rank extensions of triangular operators the problem to find the minimal rank extensions of the lower triangular part of a finite rank matrix kernel appears in a natural way see 3 in the study of minimal representations of semi separable integral operators or equivalently of minimal realizations of boundary value systems in particular one is interested in those kernels for which the minimal rank extension is unique in this paper we give a full description of this class of so called unique matrix kernels we also define and characterize lower uniqueness for finite and semi infinite block operator matrices to state one of our main results let k t s be an m x n matrix kernel defined on the square a b x a b following 3 we say that k is lower separable if the lower triangular part k of k which is defined by kl t 8 k t s a s t b admits a finite rank extension i e there exists a finite rank matrix kernel h on a b x a b such that k is the lower triangular part of h recall that for an m x n matrix kernel h on a fl x y s the rank of

history and generality of the cs decomposition abstract it is almost a quarter of a century since chandler davis and william kahan brought together the key ideas of what stewart later completed and defined to be the cs decomposition csd of a partitioned unitary matrix this paper outlines some germane points in the history of the csd pointing out the contributions of jordan of davis and kahan and of stewart and the relationship of the csd to the direct rotation of davis and kato the paper provides an easy to memorize constructive proof of the csd reviews one of its important uses and suggests a motivation for the csd which emphasizes how generally useful it is it shows the relation between the csd and generalized singular value decompositions and points out some useful nullity properties one form of the csd trivially reveals finally it shows how via the qr factorization the csd can be used to obtain interesting results for partitioned nonsingular matrices we suggest the csd be taught in its most general form with no restrictions on the two by two partition and initially with no mention of angles between subspaces

generalized inverses of morphisms with kernels abstract let x y be a morphism with kernel k x in an additive category with an involution then has a moore penrose inverse with respect to iff is invertible in this case 1 if x y then has a group inverse iff has a cokernel x k and 2 is invertible in this case 2 1

on the quantization of a self dual integrable system in this note we apply canonical quantization to the self dual particle system describing the motion of poles to a higher rank solution of the kp hierarchy explicitly determining both the quantum hamiltonian and the wavefunction it is verified that the quantum hamiltonian is trivially bispectral that is that the wavefunction can be taken to be symmetric as predicted by a widely held hypothesis of mathematical physics

more on structure ranks of matrices abstract the structure rank of a matrix is the largest rank of a submatrix which lies within a specified structure we characterize those structures avoiding the main diagonal for which a matrix and its inverse have the same structure rank thereby generalizing some results of m fiedler

minimal ranks of some quaternion matrix expressions with applications abstract suppose that p x y a bx x b cyc and q x y a bx x b cyc are quaternion matrix expressions where a is persymmetric or perskew symmetric we in this paper derive the minimal rank formula of p x y with respect to pair of matrices x and y y and the minimal rank formula of q x y with respect to pair of matrices x and y y as applications we establish some necessary and sufficient conditions for the existence of the general persymmetric or perskew symmetric solutions to some well known linear quaternion matrix equations the expressions are also given for the corresponding general solutions of the matrix equations when the solvability conditions are satisfied at the same time some useful consequences are also developed

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

rational solutions of the master symmetries of the kdv equation in this article we characterize a certain class of rational solutions of the hierarchy of master symmetries for kdv the result is that the generic rational potentials that decay at infinity and remain rational by all the flows of the master symmetry kdv hierarchy are bispectral potentials for the schrodinger operator by bispectral potentials we mean that the corresponding schrodinger operators possess families of eigenfunctions that are also eigenfunctions of a differential operator in the spectral variable this complements certain results of airault mckean moser 4 duistermaat grunbaum 10 and magri zubelli 40 as a consequence of bispectrality the rational solutions of the master symmetries turn out to be solutions of a generalized string equation

bispectral algebras of commuting ordinary differential operators we develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank n it combines and unifies the ideas of duistermaat grunbaum and wilson our construction is completely algorithmic and enables us to obtain all previously known classes or individual examples of bispectral operators the method also provides new broad families of bispectral algebras which may help to penetrate deeper into the problem

completing a symmetric 2 2 block matrix and its inverse abstract we consider the following completion problems suppose n1 n2 are nonnegative integers such that n 1 n 2 n 0 let a11 a12 a21 b22 be matrices with dimensions n1 n1 n1 n2 n2 n1 and n2 n2 respectively we determine necessary and sufficient conditions so that there exists an n2 n2 matrix a22 such that a a 11 a 21 a 12 a 22 and i a is nonsingular and symmetric and b22 is the lower right block of a partitioning of a 1 ii a is symmetric positive definite and b22 is the lower right block of a partitioning of a 1

bispectral operators of prime order the aim of this paper is to solve the bispectral problem for bispectral operators whose order is a prime number more precisely we give a complete list of such bispectral operators we use systematically the operator approach and in particular dixmier ideas on the first weyl algebra when the order is 2 the main theorem is exactly the result of duistermaat grunbaum on the other hand our proofs seem to be simpler

on moore penrose inverses of quasi kronecker structured matrices abstract the moore penrose inverse and generalized inverse of a x 1 x 2 where a x 1 x 2 are complex matrices are given under various assumptions we use the result to derive the moore penrose inverse and inverse for bdiag a k uv e with p complex matrices a k two complex p vectors u and v and a complex matrix e such block structured matrices occur in hierarchical modeling of multivariate spatial or space time gaussian processes for the latter we also give expressions of the determinant and of conditional variances

the nullity theorem for principal pivot transform abstract we generalize the nullity theorem of gustafson 1984 8 from matrix inversion to principal pivot transform several special cases of the obtained result are known in the literature such as a result concerning local complementation on graphs as an application we show that a particular matrix polynomial the so called nullity polynomial is invariant under principal pivot transform

maximal spectral distance abstract the exact upper bound for the norm distance between two normal matrices is given as a function of their eigenvalues among the applications we obtain an inequality related to the cartesian decomposition of a matrix and the evaluation of the hadamard multiplier norms for matrices in a certain broad class

relations between least squares and least rank solutions of the matrix equation axb c a matrix x is called a least squares solution of the matrix equation axb c if it minimizes the f norm of c axb a least rank solution of axb c if it minimizes the rank of c axb these two types of solution are not necessarily the same in this paper we establish necessary and sufficient conditions for the two types of solutions to coincide by using some matrix rank formulas and nested decompositions of matrices

minimal rank completions for block matrices abstract this paper deals with the following problem of completion of block matrices let aij i j be given matrices find additional matrices aij such that the completion a aij ni j 1 has lowest possible rank the structure of the set of all minimal rank completions is studied special attention is paid to the toeplitz case together with its connection to the partial realization problem

some generalized inverses of partition matrix and quotient identity of generalized schur complement abstract in this paper we first define some generalized schur complements then we study the representation of m p inverse m 1 group inverse m 1 g and drazin m 1 d for the partitioned matrix m 1 a b c d based on this we give three kind quotient identities of generalized schur complement about the partitioned matrix k a b e c d f g h l and develops the results in paper m redivo zaglia pseudo schur complement and their applications appl num math 50 2004 511 519 k jbilou a messaoudi k taba some schur complement identities and applications to matrix exatrapolation methods linear algebra appl 392 2004 195 210 in the end of the paper we give the application of these generalized schur complement in solution of linear equation and give numerical example to show our results

degrees of freedom of interference channels with comp transmission and reception we study the degrees of freedom dof of the k user interference channel with coordinated multipoint comp transmission and reception each message is jointly transmitted by m t successive transmitters and is jointly received by m r successive receivers we refer to this channel as the comp channel with a transmit cooperation order of m t and receive cooperation order of m r since the channel has a total of k transmit antennas and k receive antennas the maximum possible dof is equal to k we show that the comp channel has k dof if and only if m t m r geq k 1 the key idea is that the zero forcing of the interference corresponding to the i rm th message at the decoder of the j rm th message where j ne i can be viewed as a shared responsibility between the m t transmitters carrying the i rm th message and the m r receivers decoding the j rm th message for the general case we derive an outer bound that states that the dof is bounded above by left lceil k m t m r 2 2 right rceil for the special case with only comp transmission i e m r 1 we propose a scheme that can achieve k m t 1 2 dof for all k and conjecture that the result holds true for all k in the proposed coding scheme the m t transmitters carrying each message are used to cancel the interference introduced by this message at the first m t 1 receivers thereby allowing each of these receivers to enjoy 1 dof and asymptotic interference alignment is used to align the interfering signals at each other receiver to occupy half the signal space the achievability proofs are based on the notion of algebraic independence from algebraic geometry

fast qr iterations for unitary plus low rank matrices some fast algorithms for computing the eigenvalues of a block companion matrix have recently appeared in the literature in this paper we generalize the approach to encompass unitary plus low rank matrices of the form a u xy h where u is a general unitary matrix three important cases for applications are u unitary diagonal u unitary block hessenberg and u unitary in block cmv form our extension exploits the properties of a larger matrix hat a obtained by a certain embedding of the hessenberg reduction of a suitable to maintain its structural properties we show that hat a can be factored as product of lower and upper unitary hessenberg matrices possibly perturbed in the first k rows and moreover such a data sparse representation is well suited for the design of fast eigensolvers based on the qr iteration the resulting eigenvalue algorithm is fast and backward stable

cohomological structure in soliton equations and jacobian varieties on etudie la structure des orbites du systeme dynamique definie par la hierarchie kp totale on montre que chaque orbite est localement isomorphe a un certain groupe de cohomologie associe a une algebre commutative le systeme dynamique kp restreint a chaque orbite determine un systeme dynamique de mouvements lineaires sur lui par rapport a la structure lineaire du groupe de cohomologie

a n type dunkl operators and new spin calogero sutherland models a new family of an type dunkl operators preserving a polynomial subspace of finite dimension is constructed using a general quadratic combination of these operators and the usual dunkl operators several new families of exactly and quasi exactly solvable quantum spin calogero sutherland models are obtained these include in particular three families of quasi exactly solvable elliptic spin hamiltonians

direct approach to the band completion problem we study the problem of completion of a matrix with a specified band in such a way that the inverse be a band matrix with the same width for the general block matrix case a new approach to solve this and related problems is proposed the relations with the class of quasiseparable matrices play an important role

bispectral darboux transformations the generalized airy case abstract this paper considers darboux transformations of a bispectral operator which preserve its bispectrality a sufficient condition for this to occur is given and applied to the case of generalized airy operators of arbitrary order r 1 as a result the bispectrality of a large family of algebras of rank r is demonstrated an involution on these algebras is exhibited which exchanges the role of spatial and spectral parameters generalizing wilson s rank one bispectral involution spectral geometry and the relationship to the sato grassmannian are discussed

completion of an orthogonal projector and a related inverse mapping problem we study the completion problem for an orthogonal projector a of r n from knowledge of one or several of its columns as well as the related matrix inverse problem of finding an orthogonal projector a with ax y for two given n by k matrices x and y using geometric properties of orthogonal projections and of the singular value decomposition we derive theoretical and numerical solutions for these problems we give algorithms and test their matlab implementations

expression of the drazin and mp inverse of partitioned matrix and quotient identity of generalized schur complement in this paper we give representations of the drazin and mp inverse of a 2x2 block matrix and quotient identities for the generalized schur complement of a partitioned 3x3 matrix under conditions different than those used in recent papers on the subject we present numerical examples to illustrate our results

extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application abstract in this paper we establish the maximal and minimal ranks of the solution to the consistent system of quaternion matrix equations a 1 x c 1 a 2 x c 2 a 3 xb 3 c 3 and a 4 xb 4 c 4 which was investigated recently by wang q w wang the general solution to a system of real quaternion matrix equations comput math appl 49 2005 665 675 moreover corresponding results on some special cases are presented as an application a necessary and sufficient condition for the invariance of the rank of the general solution to the system mentioned above is presented some previous known results can be regarded as the special cases of this paper

on the structure rank of a class of structure matrices abstract we introduce a class of structure matrices called transpose complementary structure matrices we give a characterization of the n extendable transpose complementary structure matrices with trace k where k 0 2 we also prove that a nonsingular matrix and its inverse have the same s rank for this class of structure matrices s

completing a block diagonal matrix with a partially prescribed inverse abstract a complete solution of the matrix completion problem a b 1 c d is obtained in terms of solutions of a riccati type matrix equation several special cases are described in detail

extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications abstract assume that the linear quaternion matrix expression f x 1 x 2 a a 3 x 1 b 3 a 4 x 2 b 4 where x 1 x 2 are variant quaternion matrices in this paper we derive the maximal and minimal ranks of f x 1 x 2 subject to consistent systems of quaternion matrix equations a 1 x 1 c 1 x 1 b 1 c 2 and a 2 x 2 c 3 x 2 b 2 c 4 moreover corresponding results on some special cases are presented as applications we give necessary and sufficient conditions for the existence of solutions to some systems of quaternion matrix equations some previous known results can be regarded as the special cases of this paper

on the closed representation for the inverses of hessenberg matrices the general representation for the elements of the inverse of any hessenberg matrix of finite order is here extended to the reduced case with a new proof those entries are given with proper hessenbergians from the original matrix it justifies both the use of linear recurrences of unbounded order for such computations on matrices of intermediate order and some elementary properties of the inverse these results are applied on the resolvent matrix associated to a finite hessenberg matrix in standard form two examples on the unit disk are given

on disjoint range operators in a hilbert space for a bounded linear operator m in a hilbert space h various relations among the ranges r m r m r m m and the null spaces n m n m are considered from the point of view of their relations to the known classes of operators such as ep co ep weak ep gp dr or sr particular attention is paid to the range projectors of the operators m m and some further characteristics of these projectors are derived as well

how to express a parallel sum of k matrices abstract we present an alternative expression for the parallel sum of k hermitian nonnegative definite matrices by using the moore penrose inverse of a block matrix

on the characteristic polynomials and eigenvalues of some specially bordered matrices abstract using a determinant identity and leverrier s algorithm for m d system we obtain the explicit expressions of the characteristic polynomial and the eigenvalues of some specially structured bordered matrices m a u v d

an inertia formula for hermitian matrices with sparse inverses abstract let a mn be a nonsingular hermitian matrix let g be a chordal graph on vertices 1 n and suppose that g a 1 g then the inertia of a may be expressed in a simple way in terms of the inertias of those principal submatrices of a corresponding to certain readily identifiable sets of vertices of g specifically we prove that the inertia i a satisfies the identity i a c i a s m i a in which c denotes the collection of maximal cliques of g s denotes the collection of minimal vertex separators of g and m is the multiplicity of a separator

rank deficient submatrices of kronecker products of fourier matrices abstract we provide a set of maximal rank deficient submatrices of a kronecker product of two matrices a b and in particular the kronecker product of fourier matrices f f n 1 f n k we show how in the latter case maximal rank deficient submatrices can be constructed as tilings of rank one blocks several such tilings may be associated to any subgroup of the abelian group z n 1 z n k that corresponds to the matrix f the maximal rank deficient submatrices of f are also related to an uncertainty principle for fourier transforms over finite abelian groups for which we can then obtain stronger versions

the moore penrose inverses of m n block matrices and their applications abstract this article gives the expressions for the moore penrose inverses of m n block matrices when they satisfy rank additivity conditions and presents some of their special cases and applications

trace class multipliers and spectral variation of normal matrices abstract in this article we show how to estimate the trace multiplier norm of a rank 2 matrix as an application an alternative proof of a theorem of holbrook et al maximal spectral distance linear algebra appl 249 1996 197 205 on the maximal spectral distance between two normal matrices with prescribed eigenvalues is given

crum transformation and rational solutions of the non focusing nonlinear schrodinger equation a crum transformation for the linear problem of the non focusing nonlinear schrodinger nls equation is used to construct the sequence of rational solutions given in terms of tau functions expressed as double wronskian determinants the rational solutions of the adler kaup newell segur akns hierarchy have already been obtained in the work of sachs using the equivalence with the classical boussinesq hierarchy however we find that sachs description of the restrictions that must be placed on the akns tau functions in order to obtain nls rational solutions is not entirely correct we briefly comment on the constrained calogero moser systems associated with the nls rational solutions and indicate how the calogero moser equations can be obtained from a trilinear equation for the tau function which arises by reduction of the kadomtsev petviashvili kp hierarchy

inverses of 2 2 block matrices abstract in this paper the authors give explicit inverse formulae for 2 2 block matrices with three different partitions then these results are applied to obtain inverses of block triangular matrices and various structured matrices such as hamiltonian per hermitian and centro hermitian matrices

vector least squares solutions for coupled singular matrix equations the weighted least squares solutions of coupled singular matrix equations are too difficult to obtain by applying matrices decomposition in this paper a family of algorithms are applied to solve these problems based on the kronecker structures subsequently we construct a computationally efficient solutions of coupled restricted singular matrix equations furthermore the need to compute the weighted drazin and weighted moore penrose inverses and the use of tian s work and lev ari s results are due to appearance in the solutions of these problems the several special cases of these problems are also considered which includes the well known coupled sylvester matrix equations finally we recover the iterative methods to the weighted case in order to obtain the minimum d norm g vector least squares solutions for the coupled sylvester matrix equations and the results lead to the least squares solutions and invertible solutions as a special case

an explicit characterization of calogero moser systems combining theorems of halphen floquet and picard and a frobenius type analysis we characterize rational meromorphic simply periodic and elliptic kdv potentials in particular we explicitly describe the proper extension of the airault mckean moser locus associated with these three classes of algebro geometric solutions of the kdv hierarchy with special emphasis on the case of multiple collisions between the poles of solutions this solves a problem left open since the mid 1970s

on some matrix equalities for generalized inverses with applications abstract necessary and sufficient conditions are derived for the matrix equality a pn q to hold where a and n are generalized inverses of matrices some consequences and applications are also given in particular necessary and sufficient conditions are derived for the additive decompositions c a b and a a 1 a k to hold

bispectral kp solutions and linearization of calogero moser particle systems rational and soliton solutions of the kp hierarchy in the subgrassmanniangr1 are studied within the context of finite dimensional dual grassmannians in the rational case properties of the tau function which are equivalent to bispectrality of the associated wave function are identified in particular it is shown that there exists a bound on the degree of all time variables in if and only if is a rank one bispectral wave function the action of the bispectral involution in the generic rational case is determined explicitly in terms of dual grassmannian parameters using the correspondence between rational solutions and particle systems it is demonstrated that is a linearizing map of the calogero moser particle system and is essentially the map introduced by airault mckean and moser in 1977 2

the reverse order law revisited abstract necessary and sufficient conditions are given for the triple reverse order law abc c b a to hold some special cases are considered

expressions for the moore penrose inverse of block matrices involving the schur complement abstract we present a formula for the moore penrose inverse of a matrix of the form m x n y where x and y are nonsingular based on this result we develop explicit expressions for the moore penrose inverse of a 2 2 complex block matrix m we recover the cases when a schur complement in m is either equal to zero or nonsingular some applications to banded matrices and bordered matrices are indicated

upper and lower bounds for ranks of matrix expressions using generalized inverses abstract the maximal and minimal ranks of the matrix expression a 1 b 1 x 1 c 1 b 2 x 2 c 2 with respect to x 1 and x 2 are presented as applications the maximal and minimal ranks of a 1 b 1 xc 1 subject to a consistent matrix equation b 2 xc 2 a 2 are also determined in addition the maximal and minimal ranks of the schur complement d ca b with respect to the generalized inverse a of a and their various consequences are also considered

a numerical solver for general bordered tridiagonal matrix equations abstract to overcome several limitations of symbolic algorithms introduced recently for matrices of large order a fast numerical solver is proposed for the matrix linear equation a x b where the n n coefficient matrix a is a general nonsingular bordered tridiagonal matrix its sparse structure is preserved through partial givens reduction in particular the matrix inverse of a can be computed for a wide range of bordered tridiagonal linear systems a x b the solution is computed in linear time using back substitution and sherman morrison s formula numerical comparisons illustrate the results

on minimality in the partial realization problem abstract a compression algorithm is constructed which allows one to reduce an arbitrary realization of a finite sequence of matrices m 1 m r to a minimal partial realization of m 1 m r furthermore a state space criterion of minimality of a partial realization and a formula for the minimal state space dimension are obtained

solitons and almost intertwining matrices we define the algebraic variety of almost intertwining matrices to be the set of triples x y z of n n matrices for which xz yx t for a rank one matrix t a surprisingly simple formula is given for tau functions of the kp hierarchy in terms of such triples the tau functions produced in this way include the soliton and vanishing rational solutions the induced dynamics of the eigenvalues of the matrix x are considered leading in special cases to the ruijsenaars schneider particle system

on a theorem of halphen and its application to integrable systems abstract we extend halphen s theorem which characterizes solutions of certain nth order differential equations with rational coefficients and meromorphic fundamental systems to a first order n n system of differential equations as an application of this circle of ideas we consider stationary rational algebro geometric solutions of the kdv hierarchy and illustrate some of the connections with completely integrable models of the calogero moser type in particular our treatment recovers the complete characterization of the isospectral class of such rational kdv solutions in terms of a precise description of the airault mckean moser locus of their poles

the inverse of any two by two nonsingular partitioned matrix and three matrix inverse completion problems a formula for the inverse of any nonsingular matrix partitioned into two by two blocks is derived through a decomposition of the matrix itself and generalized inverses of the submatrices in the matrix the formula is then applied to three matrix inverse completion problems to obtain their complete solutions

on the existence of unimodular matrices with a prescribed submatrix abstract we study the existence of unimodular matrices with a prescribed submatrix over a ring r in particular when either r is hermite and dedekind finite or r has stable range one we give necessary and sufficient conditions for the existence of these unimodular completions

calogero moser hierarchy and kp hierarchy identification of the hierarchy of calogero moser dynamical systems with the dynamical systems of poles of certain solutions to the kp hierarchy is achieved

general expressions for the moore penrose inverse of a 2 2 block matrix abstract the moore penrose inverse of a 2 2 block matrix m m a c b d is discussed general expressions for the moore penrose inverse for the block matrix m in terms of the individual blocks a b c d are delivered without any restrictions imposed under some conditions the moore penrose inverse can be simplified the results extend earlier work by various authors

new expressions of 2 times 2 block matrix inversion and their application a 2 times 2 block matrix inversion is a tool that is frequently used in areas of control estimation theory and signal processing however one of the two diagonal entries of the block matrix should be invertible to carry out a conventional block matrix inversion in this technical note we show that this assumption can be partially released with three new types of symbolic block matrix inversion also an application example of an inverse plant model of a multi inputs and multi output mimo plant which cancels plant noise and disturbance is suggested to show the effectiveness of these new types of matrix inversion

the interplay of ranks of submatrices a banded invertible matrix t has a remarkable inverse all upper and lower submatrices of t 1 have low rank depending on the bandwidth in t the exact rank condition is known and it allows fast multiplication by full matrices that arise in the boundary element method we look for the right proof of this property of t 1 ultimately it reduces to a fact that deserves to be better known complementary submatrices of any t and t 1 have the same nullity the last figure in the paper when t is tridiagonal shows that on and above the diagonal of t 1 all rows are proportional

implicit double shift qr algorithm for companion matrices in this paper an implicit double shifted qr method for computing the eigenvalues of companion and fellow matrices will be presented companion and fellow matrices are hessenberg matrices that can be decomposed into the sum of a unitary and a rank 1 matrix the hessenberg the unitary as well as the rank 1 structures are preserved under a step of the qr method this makes these matrices suitable for the design of a fast qr method several techniques already exist for performing a qr step the implementation of these methods is highly dependent on the representation used unfortunately for most of the methods compression is needed since one is not able to maintain all three unitary hessenberg and rank 1 structures in this manuscript an implicit algorithm will be designed for performing a step of the qr method on the companion or fellow matrix based on a new representation consisting of givens transformations moreover no compression is needed as the specific representation of the involved matrices is maintained finally also a double shift version of the implicit method is presented

completing a matrix when certain entries of its inverse are specified abstract suppose a a 11 a 12 a 21 a 22 is an n n matrix where a 11 has order p q if a is nonsingular let a 1 have the transposed partitioning b 11 b 12 b 21 b 22 first we show that the nullities of a 11 and b 22 are equal the remainder of our paper deals with the following completion problem suppose m 1 m 2 n 1 n 2 are nonnegative integers such that m 1 m 2 n 1 n 2 n 0

completing triangular block matrices with maximal and minimal ranks abstract this paper considers two completion problems on block matrices with maximal and minimal ranks let a a ij be an n n block matrix where a ij n i j 1 is given and a ij 1 i is a variant block entry then determine all these variant block entries such that a a ij has maximal and minimal possible ranks respectively by making use of the theory of generalized inverses of matrices we present complete solutions to these two problems as applications we also determine maximal and minimal ranks of the matrix expression a bxc when x is a variant triangular block matrix and then present a necessary and sufficient condition for the matrix equation bxc a to have a triangular block solution

representations of generalized inverses of partitioned matrix involving schur complement in this article we consider some representations of 1 3 1 4 1 2 3 and 1 2 4 inverses of the partitioned matrix m which are equivalent to some rank additivity conditions we present the application of these results to generalizations of the sherman morrison woodbury type formulae

complementary schur complements abstract for square matrices the relationship is discussed between the notion of schur complement and the equivalent concepts of matricial coupling and equivalence after extension special attention is paid to the hermitian and skew hermitian cases