minors of the moore penrose inverse abstract let q k n 1 k 1 1 k n denote the strictly increasing sequences of k elements from 1 n for q k n we denote by a the submatrix of a with rows indexed by columns by the submatrix obtained by deleting the rows and columns is denoted by a lsqb rsqb for nonsingular a r n n the jacobi identity relates the minors of the inverse a 1 to those of a det a 1 lsqb rsqb 1 k i 1 i k i 1 i det a lsqb rsqb det a for any q k n we generalize the jacobi identity to matrices a r m n r expressing the minors of the moore penrose inverse a in terms of the minors of the maximal nonsingular submatrices a ij of a in our notation det a lsqb rsqb 1 vol 2 a i j n det a ij a a ij for any q k m q k n 1 k r where vol 2 a denotes the sum of squares of determinants of r r submatrices of a we apply our results to questions concerning the nonnegativity of principal minors of the moore penrose inverse

a geometric property of the least squares solution of linear equations abstract we derive an explicit determinantal formula for the least squares solution of an overdetermined system of linear equations from this formula it follows that the least squares solution lies in the convex hull of the solutions to the square subsystems of the original system we extend this result and prove that this geometric property holds for a more general class of problems which includes the weighted least squares and lp norm minimization problems

generalized inverses with proportional minors let a h be matrices of rank r and of order m n and n m respectively over an integral domain it is shown that a admits a g inverse whose r r minors are propotional to the corresponding minors of h if and only if the trace of the rth compound of ah is invertible we also obtain a determinantal formula for such a g inverse when it exists the results generalize earlier work on the existence of the moore penrose and group inverses

stable embedding of grassmann manifold via gaussian random matrices compressive sensing cs provides a new perspective for data reduction without compromising performance when the signal of interest is sparse or has intrinsically low dimensional structure the theoretical foundation for most of the existing studies on cs is based on the stable embedding i e a distance preserving property of vectors that are sparse or in a union of subspaces via random measurement matrices to the best of our knowledge few existing literatures of cs have clearly discussed the stable embedding of linear subspaces via compressive measurement systems in this paper we explore a volume based stable embedding of multidimensional signals based on grassmann manifold via gaussian random measurement matrices the grassmann manifold is a topological space in which each point is a linear vector subspace and is widely regarded as an ideal model for multidimensional signals generated from linear subspaces in this paper we formulate the linear subspace spanned by multidimensional signal vectors as points on the grassmann manifold and use the volume and the product of sines of principal angles also known as the product of principal sines as the generalized norm and distance measure for the space of grassmann manifold we prove a volume preserving embedding property for points on the grassmann manifold via gaussian random measurement matrices i e the volumes of all parallelotopes from a finite set in grassmann manifold are preserved upon compression this volume preserving embedding property is a multidimensional generalization of the conventional stable embedding properties which only concern the approximate preservation of lengths of vectors in certain unions of subspaces in addition we use the volume preserving embedding property to explore the stable embedding effect on a generalized distance measure of grassmann manifold induced from volume it is proved that the generalized distance measure i e the product of principal sines between different points on the grassmann manifold is well preserved in the compressed domain via gaussian random measurement matrices numerical simulations are also provided for validation

recurrent neural networks for computing pseudoinverses of rank deficient matrices three recurrent neural networks are presented for computing the pseudoinverses of rank deficient matrices the first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of moore penrose inversion under the condition of zero initial states the second recurrent neural network consists of an array of neurons corresponding to a pseudoinverse matrix with decaying self connections and constant connections in each row or column the third recurrent neural network consists of two layers of neuron arrays corresponding respectively to a pseudoinverse matrix and a lagrangian matrix with constant connections all three recurrent neural networks are also composed of a number of independent subnetworks corresponding to the rows or columns of a pseudoinverse the proposed recurrent neural networks are shown to be capable of computing the pseudoinverses of rank deficient matrices

three results in connection with inverse matrices abstract first we show that the moore penrose solution of an arbitrary system of linear equations is a convex combination of the solutions of all uniquely solvable partial systems the other two results concern the elements of inverse toeplitz band matrices namely the asymptotic behavior of a determinant appearing in a formula of d s meek and a modification of a formula of w d hoskins and p j ponzo for matrices with binomial coefficients in the limit case

large deviation probabilities and dominating points for open convex sets nonlogarithmic behavior the existence of a dominating point for an open convex set and a corresponding representation formula for large deviation probabilities are established in the infinite dimensional setting under conditions which are both necessary and sufficient and follow from those used previously in mathbb r d a precise nonlogarithmic estimate of large deviation probabilities applicable to gaussian measures is also included

a best upper bound for the 2 norm condition number of a matrix abstract let a be an n n nonsingular real or complex matrix the best possible upper bound for the ratio of the largest and smallest singular values of a using tr a a det a and n only is obtained a comparison with an earlier bound is given and the singular and nonsquare cases are included if all the eigenvalues of a are real and positive the best possible upper bound for the ratio of the largest and smallest eigenvalues of a involving tr a det a and n only is presented as well

a linear receiver for direct sequence spread spectrum multiple access systems with antenna arrays and blind adaptation a linear receiver for direct sequence spread spectrum multiple access communication systems under the aperiodic random sequence model is considered the receiver consists of the conventional matched filter followed by a tapped delay line with the provision of incorporating the use of antenna arrays it has the ability of suppressing multiple access interference mai and narrowband interference in some weighted proportions as well as combining multipath components without explicit estimation of any channel conditions under some specific simplified channel models the receiver reduces to the minimum variance distortionless response beamformer the rake receiver a notch filter or an mai suppressor the interference rejection capability is made possible through a suitable choice of weights in the tapped delay line the optimal weights can be obtained by straightforward but computationally complex eigenanalysis in order to reduce the computational complexity a simple blind adaptive algorithm is also developed

recurrent neural networks for computing weighted moore penrose inverse1this project is supported by the national nature science foundation of china doctoral point foundation of china and youth science foundation of universities in shanghai of china 1 three recurrent neural networks are presented for computing the weighted moore penrose inverse of rank deficient matrices the first recurrent neural network has the dynamical equation similar to the one proposed earlier for matrix inversion and is capable of weighted moore penrose inverse under the condition of zero initial states the second recurrent neural network consists of an array of neurons corresponding to a weighted moore penrose inverse matrix with decaying self connections and constant connections in each row or column the third recurrent neural network consists of two layers of neuron arrays corresponding respectively to a weighted moore penrose inverse and a lagrangian matrix with constant connections

a berry esseen bound for symmetric statistics the rate of convergence of the distribution function of a symmetric function of n independent and identically distributed random variables to its normal limit is investigated under appropriate moment conditions the rate is shown to be o left n frac 1 2 right this theorem generalizes many known results for special cases and two examples are given possible further extensions are indicated

product cosines of angles between subspaces abstract let cos l m i 1 cos i denote the product of the cosines of the principal angles i between the subspaces l and m the direction cosines of an r dimensional subspace l are the n r numbers cos l r n j j q r n where q r n the set of increasing sequences of r elements from 1 n and r n j x x k r n x k 0 for k j the basic decomposition of a linear operator a r n r m with rank a r 0 is a i i a j j a cos 2 r a r m l cos 2 r a t r n j b ij a convex combination of nonsingular linear operators b ij r n j r n i here j a i q r m rank a i r and j a j q r n rank a j r the product cosines are related to the matrix volume defined as the product of its nonzero singular values the moore penrose inverse a is characterized as having the minimal volume among all 1 2 inverses of a indeed if g is a 1 2 inverse of a with range r g t and null space n g s then vol g vol a cos t r a t cos s n a t

adaptive generalized eigen pairs extraction algorithms and their convergence analysis the generalized hermitian eigenvalue problem ghep is of great use in modern signal processing compared with other methods neural network model based algorithms provide an efficient way to solve such problems online up to now a class of neural network model based generalized feature extraction algorithms have been reported in the literature however most existing noncoupled algorithms suffer from the so called speed stability problem in contrast coupled learning algorithms estimate the generalized eigenvector and generalized eigenvalue simultaneously in coupled manners therefore they can solve the speed stability problem and perform better than noncoupled algorithms in this paper we first propose a coupled generalized system which is obtained by using the newton s method and a novel generalized information criterion then based on this coupled generalized system we successfully obtain two coupled algorithms with normalization steps for minor principal generalized eigen pairs extraction and the technique of multiple generalized eigen pairs extraction is also introduced in this paper the convergence of the proposed algorithms is justified by deterministic discrete time ddt approach with the efficiency and feasibility validated by experimental results as well compared with similar work the proposed algorithms have lower computational complexity better numerical stability and higher estimation accuracy

adaptive normalized quasi newton algorithms for extraction of generalized eigen pairs and their convergence analysis the main contributions of this paper are to propose and analyze fast and numerically stable adaptive algorithms for the generalized hermitian eigenvalue problem ghep which arises in many signal processing applications first for given explicit knowledge of a matrix pencil we formulate two novel deterministic discrete time ddt systems for estimating the generalized eigen pair eigenvector and eigenvalue corresponding to the largest smallest generalized eigenvalue by characterizing a generalized eigen pair as a stationary point of a certain function the proposed ddt systems can be interpreted as natural combinations of the normalization and quasi newton steps for finding the solution second we present adaptive algorithms corresponding to the proposed ddt systems moreover we establish rigorous analysis showing that for a step size within a certain range the sequence generated by the ddt systems converges to the orthogonal projection of the initial estimate onto the generalized eigensubspace corresponding to the largest smallest generalized eigenvalue numerical examples demonstrate the practical applicability and efficacy of the proposed adaptive algorithms

a special irreducible matrix representation of the real clifford algebra c 3 1 4 4 dirac gamma matrices irreducible matrix representations of the clifford algebras c 3 1 c 1 3 c 4 0 are an essential part of many calculations in quantum physics although the final physical results do not depend on the applied representation of the dirac matrices e g due to the invariance of traces of products of dirac matrices the appropriate choice of the representation used may facilitate the analysis the present paper introduces a particularly symmetric real representation of 4 4 dirac matrices majorana representation which may prove useful in the future as a by product a compact formula for transformed pauli matrices is found the consideration is based on the role played by isoclinic 2 planes in the geometry of the real clifford algebra c 3 0 which provide an invariant geometric frame for it it can be generalized to larger clifford algebras

moderate deviation probabilities for open convex sets nonlogarithmic behavior precise asymptotics for moderate deviation probabilities are established for open convex sets in both the finite and infinite dimensional settings our results are based on the existence of dominating points for these sets a related representation formula and asymptotics for the integral term in this formula

coupled principal component analysis a framework for a class of coupled principal component learning rules is presented in coupled rules eigenvectors and eigenvalues of a covariance matrix are simultaneously estimated in coupled equations coupled rules can mitigate the stability speed problem affecting noncoupled learning rules since the convergence speed in all eigendirections of the jacobian becomes widely independent of the eigenvalues of the covariance matrix a number of coupled learning rule systems for principal component analysis two of them new is derived by applying newton s method to an information criterion the relations to other systems of this class the adaptive learning algorithm ala the robust recursive least squares algorithm rrlsa and a rule with explicit renormalization of the weight vector length are established

an algorithm for the symmetric generalized eigenvalue problem abstract a new method is presented for the solution of the matrix eigenvalue problem ax bx where a and b are real symmetric square matrices and b is positive semidefinite it reduces a and b to diagonal form by congruence transformations that preserve the symmetry of the problem this method is closely related to the qr algorithm for real symmetric matrices

the classical adjoint abstract this paper summarizes the historical background of the notion of the classical adjoint as outlined by muir and provides applications of the adjoint to various studies of generalized invertibility of matrices over commutative rings specifically in this setting the classical adjoint is used to provide a novel proof of von neumann s 1936 observation that every matrix over a regular ring is regular and to provide a necessary and sufficient condition for the existence of the moore penrose inverse of a given matrix in particular a representation of the moore penrose inverse is given that leads to an immediate proof of moore s 1920 formula specifying the entries of his reciprocal in terms of determinants

the representation and approximation for the generalized inverse at s 2 we present a unified representation theorem for the generalized inverses a t s 2 which are 2 or outer inverses having the prescribed range t and null space s specific expressions and computational procedures for the generalized inverses a t s 2 can be uniformly derived

a volume associated with m x n matrices abstract let a mxnr with nonzero singular values 1 2 r the volume of a vol a is defined as zero if r 0 and otherwise vola ri 1 i or equivalently vola det2aij summing over all r x r nonsingularsubmatrices aij the matrix volume vol a generalizes the absolute value of the determinant from nonsingular to arbitrary matrices any r dimensional unit cube in r at is mapped under a into a parallelepiped of volume vol a this paper gives properties and applications of vol a in particular the moore penrose inverse of a is a convex combination of ordinary inverses of its maximal nonsingular submatrices a i j ij or is a convex combination of moore penrose inverses of its maximal full rank submatrices a i i j j where the convex weights are proport ional to the squares of volumes of the corresponding submatrices since a a the matrix a is a convex combination of inverses of submatrices of a interestingly the weights of these convex combinations are the above weights in transposed position

stability results and strong invariance principles for partial sums of banach space valued random variables a general stability theorem for b valued random variables is obtained which refines a result of kuelbs and zinn our proof is based on two exponential inequalities for sums of independent b valued r v s essentially due to yurinskii and appears particularly simple we then use our theorem to prove strong invariance principles lil results and other related stability results for sums of i i d b valued r v s in the domain of attraction of a gaussian law most of these results seem to be still unknown for real valued r v s

dominating points and large deviations for random vectors we establish a representation formula useful for obtaining precise large deviation probabilities for convex open subsets of a banach space these estimates are based on the existence of dominating points in this setting

a quasi newton adaptive algorithm for generalized symmetric eigenvalue problem we first recast the generalized symmetric eigenvalue problem where the underlying matrix pencil consists of symmetric positive definite matrices into an unconstrained minimization problem by constructing an appropriate cost function we then extend it to the case of multiple eigen vectors using an inflation technique based on this asymptotic formulation we derive a quasi newton based adaptive algorithm for estimating the required generalized eigen vectors in the data case the resulting algorithm is modular and parallel and it is globally convergent with probability one we also analyze the effect of inexact inflation on the convergence of this algorithm and that of inexact knowledge of one of the matrices in the pencil on the resulting eigenstructure simulation results demonstrate that the performance of this algorithm is almost identical to that of the rank one updating algorithm of karasalo 1986 further the performance of the proposed algorithm has been found to remain stable even over 1 million updates without suffering from any error accumulation problems

a novel unified and self stabilizing algorithm for generalized eigenpairs extraction generalized eigendecomposition problem has been widely employed in many signal processing applications in this paper we propose a unified and self stabilizing algorithm which is able to extract the first principal and minor generalized eigenvectors of a matrix pencil of two vector sequences adaptively furthermore we extend the proposed algorithm to extract multiple generalized eigenvectors the performance analysis shows that only the desired equilibrium point of the proposed algorithm is stable and all others are unstable repellers or saddle points convergence analysis based on the deterministic discrete time approach shows that for a step size within a certain range the norm of the principal minor state vector converges to a fixed value that relates to the corresponding principal minor generalized eigenvalue thus the proposed algorithm is a generalized eigenpairs eigenvectors and eigenvalues extraction algorithm finally the simulation experiments are carried to further demonstrate the efficiency of the proposed algorithm

robust adaptive beamforming for general rank signal models the performance of adaptive beamforming methods is known to degrade severely in the presence of even small mismatches between the actual and presumed array responses to the desired signal such mismatches may frequently occur in practical situations because of violation of underlying assumptions on the environment sources or sensor array this is especially true when the desired signal components are present in the beamformer training data snapshots because in this case the adaptive array performance is very sensitive to array and model imperfections the similar phenomenon of performance degradation can occur even when the array response to the desired signal is known exactly but the training sample size is small we propose a new powerful approach to robust adaptive beamforming in the presence of unknown arbitrary type mismatches of the desired signal array response our approach is developed for the most general case of an arbitrary dimension of the desired signal subspace and is applicable to both the rank one point source and higher rank scattered source fluctuating wavefront desired signal models the proposed robust adaptive beamformers are based on explicit modeling of uncertainties in the desired signal array response and data covariance matrix as well as worst case performance optimization simple closed form solutions to the considered robust adaptive beamforming problems are derived our new beamformers have a computational complexity comparable with that of the traditional adaptive beamforming algorithms while at the same time offer a significantly improved robustness and faster convergence rates

minimal properties of moore penrose inverses abstract let ax y be consistent let x0 gy be any minimum norm solution satisfying ag ag and let a be the moore penrose inverse of a it is shown that g a for any in a class containing the unitarily invariant matrix norms the conditioning of the system ax y is studied via condition numbers c a g it is shown that c a g c a a for every moreover bounds on c a g are given in terms of singular values parallel results are found when a and g are symmetric with applications to linear models of less than full rank

a novel iterative method for computing generalized inverse in this letter we propose a novel iterative method for computing generalized inverse based on a novel kkt formulation the proposed iterative algorithm requires making four matrix and vector multiplications at each iteration and thus has low computational complexity the proposed method is proved to be globally convergent without any condition furthermore for fast computing generalized inverse we present an acceleration scheme based on the proposed iterative method the global convergence of the proposed acceleration algorithm is also proved finally the effectiveness of the proposed iterative algorithm is evaluated numerically

neural network for computing pseudoinverses and outer inverses of complex valued matrices we propose two continuous time neural networks for computing generalized inverses of complex valued matrices with rank deficient cases the first of them is applicable in the pseudoinverse computation and the second one is applicable in construction of outer inverses the proposed continuous time neural networks have a low complexity of implementation and they are proved to be globally convergent without any condition compared with the existing algorithms for computing the pseudoinverse and outer inverses of matrices the global convergence of the proposed continuous time neural networks is analyzed in the complex domain effectiveness of the proposed continuous time neural networks is evaluated numerically via examples

rls based adaptive algorithms for generalized eigen decomposition the aim of this paper is to develop efficient online adaptive algorithms for the generalized eigen decomposition problem which arises in a variety of modern signal processing applications first we reinterpret the generalized eigen decomposition problem as an unconstrained minimization problem by constructing a novel cost function second by applying projection approximation method and recursive least square rls technique to the cost function a parallel adaptive algorithm for a basis for the r dimensional r 0 dominant generalized eigen subspace and a sequential algorithm based on deflation technique for the first r dominant generalized eigenvectors are derived these algorithms can be viewed as counterparts of the extended projection approximation subspace tracking past and pastd algorithms respectively furthermore we modify the parallel algorithm to explicitly estimate the first r generalized eigenvectors in parallel not the generalized eigen subspace more important the modified parallel algorithm can be used to extract multiple generalized eigenvectors of two nonstationary sequences while the proposed sequential algorithm lacks this ability because of slow convergence of minor generalized eigenvectors due to error propagation of the deflation technique third following convergence analysis methods for past and pastd we prove the asymptotic convergence properties of the proposed algorithms finally computer simulations are performed to investigate the accuracy and the speed advantages of the proposed algorithms

on principal angles between subspaces in rn abstract let l m be subspaces in r n dim l l dim m m then the principal angles between l and m 0 1 2 l 2 are given by cos i x i y i x i y i max x y x y x l y m x x k y y k k 1 1 where x i y i l m i 1 l are the corresponding pairs of principal vectors we also define sin l m l i 1 sin i cos l m l i 1 cos i we study relations between the principal angles and the volume of a matrix a r m n defined by vol a det 2 a ij summing over all r r submatrices a ij of a sample results are the following generalizations of the hadamard and cauchy schwarz inequalities 1 let a a 1 a 2 a 1 r n n 1 l a 2 r n n 2 m rank a l m then vol a vol a 1 vol a 2 sin r a 1 r a 2 2 let b c r n r r then det b t c volbvolccos r b r c

the geometry of basic approximate and minimum norm solutions of linear equations the basic solutions of the linear equations ax b are the solutions of subsystems corresponding to maximal nonsingular submatrices of a the convex hull of the basic solutions is denoted by c c a b given 1 p the lp approximate solutions of ax b denoted x p are minimizers of ax b p given m dm the set of positive diagonal m m matrices the solutions of minx m ax b p are called scaledlp approximate solutions for 1 p1 p2 the minimum lp2 norm lp1 approximate solutions are denoted x p1 p2 main results 1 1 if a rm nm then c contains all some minimum lp norm solutions for 1 p p 2 2 for general a and any 1 p1 p2 the set c contains all x p1 p2 3 3 the set of scaled lp approximate solutions with m ranging over dm is the same for all 1 p 4 4 the set of scaled least squares solutions has the same closure as the set of solutions of minx f ax b where f rm r ranges over all strictly isotone functions

new proofs of two representations and minor of generalized inverse a 2 t s abstract in this paper we prove the limit representation for the generalized inverse a t s 2 of a given complex matrix a in a simpler way then using this representation two known results the generalized inverse a t s 2 as an analogs of convex combination with the minor of adja and the representation of the minor of the generalized inverse a t s 2 were derived anew as a special case of the minor a determinantal representation of the generalized inverse a t s 2 is obtained

projection approximation subspace tracking subspace estimation plays an important role in a variety of modern signal processing applications we present a new approach for tracking the signal subspace recursively it is based on a novel interpretation of the signal subspace as the solution of a projection like unconstrained minimization problem we show that recursive least squares techniques can be applied to solve this problem by making an appropriate projection approximation the resulting algorithms have a computational complexity of o nr where n is the input vector dimension and r is the number of desired eigencomponents simulation results demonstrate that the tracking capability of these algorithms is similar to and in some cases more robust than the computationally expensive batch eigenvalue decomposition relations of the new algorithms to other subspace tracking methods and numerical issues are also discussed