generalization of a fundamental matrix abstract it is shown that for a finite ergodic markov chain basic descriptive quantities such as the stationary vector and mean first passage matrix may be calculated using any one of a class of fundamental matrices new applications of the use of these operators are discussed

tracking while zooming using affine transfer and multifocal tensors this paper presents algorithms for tracking unknown objects in the presence of zoom since prior models are unavailable point and line matches in affine views are used to characterize the structure and to transfer a fixation point into new images in a sequence because any affine projection matrix is permitted the intrinsic camera parameters such as focal length may change freely also since the techniques do not require long feature tracks a further desirable property is insensitivity to partial occlusion caused for instance by part of the object falling off the image plane while zooming in if only point matches are available a previous method based on factorization is applied when also incorporating lines the affine trifocal and quadrifocal tensors are used for tracking in monocular and stereo systems respectively methods for computing the tensors minimizing algebraic error are developed in comparison with their projective counterparts the affine tensors offer significant advantages in terms of computation time and convenience of parameterization and the relations between the different tensors are shown to be much simpler successful tracking is demonstrated on several real image sequences

twist based acquisition and tracking of animal and human kinematics this paper demonstrates a new visual motion estimation technique that is able to recover high degree of freedom articulated human body configurations in complex video sequences we introduce the use and integration of a mathematical technique the product of exponential maps and twist motions into a differential motion estimation this results in solving simple linear systems and enables us to recover robustly the kinematic degrees of freedom in noise and complex self occluded configurations a new factorization technique lets us also recover the kinematic chain model itself we are able to track several human walk cycles several wallaby hop cycles and two walk cycels of the famous movements of eadweard muybridge s motion studies from the last century to the best of our knowledge this is the first computer vision based system that is able to process such challenging footage

mixed type reverse order laws for 1 3 4 generalized inverses over hilbert spaces abstract the reverse order laws for 1 3 4 generalized inverses of a product of two operators have been studied by wang et al j wang h zhang g ji a generalized reverse order law for the products of two operators journal of shaanxi normal university 38 4 2010 13 17 in this paper using a block operator matrix technique we study mixed type reverse order laws for 1 3 4 generalized inverses over hilbert spaces

reverse order law for reflexive generalized inverses of products of matrices abstract the relationships between generalized inverses of the product of two matrices a b and the product of generalized inverses of a b have been studied in the literature by applying the product singular value decomposition p svd in this paper we derive equivalent conditions for b 1 2 a 1 2 ab 1 2 ab 1 2 b 1 2 a 1 2 b 1 2 a 1 2 ab 1 2 and equivalent conditions for ab 1 b 1 a 1

dynamic 3d models with local and global deformations deformable superquadrics the authors present a physically based approach to fitting complex three dimensional shapes using a novel class of dynamic models that can deform both locally and globally they formulate the deformable superquadrics which incorporate the global shape parameters of a conventional superellipsoid with the local degrees of freedom of a spline the model s six global deformational degrees of freedom capture gross shape features from visual data and provide salient part descriptors for efficient indexing into a database of stored models the local deformation parameters reconstruct the details of complex shapes that the global abstraction misses the equations of motion which govern the behavior of deformable superquadrics make them responsive to externally applied forces the authors fit models to visual data by transforming the data into forces and simulating the equations of motion through time to adjust the translational rotational and deformational degrees of freedom of the models model fitting experiments involving 2d monocular image data and 3d range data are presented

some results on partial ordering and reverse order law of elements of c algebras abstract in this paper we establish some results relating star left star right star minus ordering and the reverse order law under certain conditions on moore penrose invertible elements of c algebras

further results on the reverse order law for the group inverse in rings in this paper we use the drazin inverse to derive some new equivalences of the reverse order law for the group inverse in unitary rings moreover if the ring has an involution we present more equivalences when both involved elements are ep

symmetries of the car algebra a automorphism of period two of the uhf c algebra of type 2 is constructed for which the fixed point algebra is not approximately finite dimensional the construction also yields a diagonal maximal commutative c subalgebra whose spectrum is not totally disconnected variations and consequences of the construction are discussed

c ondensation conditional density propagation forvisual tracking the problem of tracking curves in dense visual clutter is challenging kalman filtering is inadequate because it is based on gaussian densities which being unimo dal cannot represent simultaneous alternative hypotheses the condensation algorithm uses factored sampling previously applied to the interpretation of static images in which the probability distribution of possible interpretations is represented by a randomly generated set condensation uses learned dynamical models together with visual observations to propagate the random set over time the result is highly robust tracking of agile motion notwithstanding the use of stochastic methods the algorithm runs in near real time

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

generalized inverses of markovian kernels in terms of properties of the markov chain abstract all one condition generalized inverses of the markovian kernel i p where p is the transition matrix of a finite irreducible markov chain can be uniquely specified in terms of the stationary probabilities and the mean first passage times of the underlying markov chain special sub families include the group inverse of i p kemeny and snell s fundamental matrix of the markov chain and the moore penrose g inverse the elements of some sub families of the generalized inverses can also be re expressed involving the second moments of the recurrence time variables some applications to kemeny s constant and perturbations of markov chains are also considered

the generalized moore penrose inverse abstract we define the generalized moore penrose inverse and give necessary and sufficient conditions for its existence over an integral domain we also prove its uniqueness and give a formula for it which leads us towards a generalized cramer s rule to find the generalized moore penrose solution

factorization of weighted ep elements in c algebras abstract we present characterizations of weighted ep elements in c algebras using different kinds of factorizations

object tracking in image sequences using point features this paper presents an object tracking technique based on the bayesian multiple hypothesis tracking mht approach two algorithms both based on the mht technique are combined to generate an object tracker the first mht algorithm is employed for contour segmentation the segmentation of contours is based on an edge map the segmented contours are then merged to form recognisable objects the second mht algorithm is used in the temporal tracking of a selected object from the initial frame an object is represented by key feature points that are extracted from it the key points mostly corner points are detected using information obtained from the edge map these key points are then tracked through the sequence to confirm the correctness of the tracked key points the location of the key points on the trajectory are verified against the segmented object identified in each frame if an acceptable number of key points lie on or near the contour of the object in a particular frame n th frame we conclude that the selected object has been tracked identified successfully in frame n

contour tracking with automatic motion model switching abstract in this paper we present an efficient contour tracking algorithm which can track 2d silhouette of objects in extended image sequences we demonstrate the ability of the tracker by tracking highly deformable contours such as walking people captured by a static camera we represent contours silhouette of moving objects by using a cubic b spline the tracking algorithm is based on tracking a lower dimensional shape space as opposed to tracking in spline space tracking the lower dimensional space has proved to be fast and efficient the tracker is also coupled with an automatic motion model switching algorithm which makes the tracker robust and reliable when the object of interest is moving with multiple motion the model based tracking technique provided is capable of tracking rigid and non rigid object contours with good tracking accuracy

some stochastic properties of semi magic and magic markov chains abstract gustafson and styan gustafson and styan superstochastic matrices and magic markov chains linear algebra appl 430 2009 2705 2715 examined the mathematical properties of superstochastic matrices the transition matrices of magic markov chains formed from scaled magic squares this paper explores the main stochastic properties of such chains as well as semi magic chains with doubly stochastic transition matrices stationary distribution generalized inverses of markovian kernels mean first passage times variances of the first passage times and expected times to mixing are considered some general results are developed some observations from the chains generated by matlab are discussed some conjectures are presented and some special cases involving three and four states are explored in detail

learning and classification of complex dynamics standard exact techniques based on likelihood maximization are available for learning auto regressive process models of dynamical processes the uncertainty of observations obtained from real sensors means that dynamics can be observed only approximately learning can still be achieved via em k expectation maximization em based on kalman filtering this cannot handle more complex dynamics however involving multiple classes of motion a problem arises also in the case of dynamical processes observed visually background clutter arising for example in camouflage produces non gaussian observation noise even with a single dynamical class non gaussian observations put the learning problem beyond the scope of em k for those cases we show here how em c based on the condensation algorithm which propagates random particle sets can solve the learning problem here learning in clutter is studied experimentally using visual observations of a hand moving over a desktop the resulting learned dynamical model is shown to have considerable predictive value when used as a prior for estimation of motion the burden of computation in visual observation is significantly reduced multiclass dynamics are studied via visually observed juggling plausible dynamical models have been found to emerge from the learning process and accurate classification of motion has resulted in practice em c learning is computationally burdensome and the paper concludes with some discussion of computational complexity

further results on the reverse order law for the moore penrose inverse in rings with involution abstract we present some equivalent conditions of the reverse order law for the moore penrose inverse in rings with involution extending some well known results to more general settings then we apply this result to obtain a set of equivalent conditions to the reverse order rule for the weighted moore penrose inverse in c algebras

diagonality inc algebras soit a une c algebre et a sa les elements hermitiens soit i un ideal ferme separable dans une c algebre a et on suppose que les elements de a sa de spectres finis sont denses dans a sa si a a sa et e 0 alors il existe b a sa diagonal relativement a i et il existe x i sa tel que x e et a b x

reverse order laws in c algebras abstract in this paper we offer purely algebraic necessary and sufficient conditions for reverse order laws for generalized inverses in c algebras extending rank conditions for matrices and range conditions for hilbert space operators

new characterizations of ep generalized normal and generalized hermitian elements in rings abstract we present a number of new characterizations of ep elements in rings with involution in purely algebraic terms then we study equivalent conditions for an element a in a ring with involution to satisfy a n a a a n or a n a n for arbitrary n n for n 1 we present some new characterizations of normal and hermitian elements in rings with involution

group moore penrose core and dual core inverse in rings with involution abstract let r be a ring with involution the recently introduced notions of the core and dual core inverse are extended from matrix to an arbitrary ring case it is shown that the group moore penrose core and dual core inverse are closely related and they can be treated in the same manner using appropriate idempotents the several characterizations of these inverses are given some new properties are obtained and some known results are generalized a number of characterizations of ep elements in r are obtained it is shown that core and dual core inverse belong to the class of inverses along an element and to the class of b c inverses

c algebras of real rank zero abstract the concept of real rank of a c algebra is introduced as a non commutative analogue of dimension it is shown that real rank zero is equivalent to the previously defined conditions fs and hp and that it is invariant under strong morita equivalence in particular under stable isomorphism real rank zero is also invariant under inductive limits and split extensions and the class may well be regarded as the conceptual completion of the af algebras in some cases notably when the algebra is matroid it is shown that the multiplier algebra also has real rank zero although that is not true in general by a result of g j murphy this implies a weyl von neumann type result for self adjoint multiplier elements in these cases

generalized inverses and their application to applied probability problems the main aim of this paper is to examine the applicability of generalized inverses to a wide variety of problems in applied probability where a markov chain is present either directly or indirectly through some form of imbedding by characterizing all generalized inverses of i p where p is the transition matrix of a finite irreducible discrete time markov chain we are able to obtain general procedures for finding stationary distributions moments of the first passage time distributions and asymp totic forms for the moments of the occupation time random variables it is shown that all known explicit methods for examining these problems can be expressed in this generalized inverse framework more generally in the context of a markov renewal process setting the aforementioned problems are also examined using generalized inverses of i p as a special case markov chains in continuous time are considered and we show that the generalized inverse technique can be applied directly to the infinitesimal generator of the process instead of to i p where p is the transition matrix of the discrete time jump markov chain

on the structure of simple c algebras tensored with a uhf algebra ii let d be a simple unital c algebra let b be a uhf algebra and put a b d it is proved that if p q a are projections and p q for all quasitraces on a then p q in the sense of murray and von neumann a more general result involving positive operators in a is also proved if d has finitely many extremal quasi traces and the projections in d k separate these then it is proved that a has real rank zero finally it is proved that provided d is stably finite then each positive state on k0 d lifts to a quasi trace

dimension and stable rank in the k theory of c algebras in topological k theory which can be viewed as the algebraic side of the theory of vector bundles some of the interesting properties which one investigates are for example the conditions under which bundles must possess trivial direct summands or the extent to which the cancellation property for direct whitney sums of bundles holds such properties turn out to be controlled in part by the dimension of the base space and results describing the nature of this control are among what are frequently called stability results 17 during the past few years there has been a sudden flowering of a theory for c algebras see surveys 35 36 37 since c algebras are profitably thought of as non commutative locally compact spaces with the finitely generated projective modules being the appropriate generalization of vector bundles according to swan s theorem 40 it would be natural to look for stability results for c algebras but until now the theory has been focussed on the c groups of c algebras and there has been little discussion of stability properties presumably in part for lack of an appropriate concept of dimension for c algebras one of the main objectives of this paper is to introduce a concept of dimension for c algebras which directly generalizes the classical concept of dimension for compact spaces 16 20 22 and to develop some techniques for calculating it notably for c algebras which are obtained as crossed product algebras for an action of the group of integers in algebraic k theory which is the generalization of topological c theory to rings a substantial number of stability properties have been discovered 2 39 42 48 within that theory the concept which has played a role most analogous to that of dimension is the bass stable rank but despite its many successes the bass stable rank has often been difficult to calculate even for relatively uncomplicated rings another main objective of the present paper is to make accessible for application to c algebras some of the stability results from algebraic c theory this comes about from the fact that the concept of dimension which we introduce dominates the bass stable rank i was led to seek stability results for c algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation c algebras discussed in 30 which as elliott 11 has suggested can appropriately be called 2 dimensional non commutative tori in more concrete terms the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent the present paper provides the general theory which is needed for the affirmative answer which i obtained in a subsequent paper i will derive the

the absorption laws for the generalized inverses abstract in this paper we give necessary and sufficient conditions for the absorption laws in terms of 1 1 2 1 3 and 1 4 inverses also we consider the various types of mixed absorption law for the generalized inverses

effective tracking through tree search a new contour tracking algorithm is presented tracking is posed as a matching problem between curves constructed out of edges in the image and some shape space describing the class of objects of interest the main contributions of the paper are to present an algorithm which solves this problem accurately and efficiently in a provable manner in particular the algorithm s efficiency derives from a novel tree search algorithm through the shape space which allows for much of the shape space to be explored with very little effort this latter property makes the algorithm effective in highly cluttered scenes as is demonstrated in an experimental comparison with a condensation tracker

weighted partial isometries and weighted ep elements in c algebras we investigate weighted partial isometries weighted ep elements weighted star dagger weighted normal and weighted hermitian elements of c algebras

moore penrose inverse in rings with involution abstract we study the moore penrose inverse mp inverse in the setting of rings with involution the results include the relation between regular mp invertible and well supported elements we present an algebraic proof of the reverse order rule for the mp inverse valid under certain conditions on mp invertible elements applications to c algebras are given

when is b a a generalized inverse of ab in practice factorizations of a generalized inverse often arise from factorizations of the matrix which is to be inverted in addition to full rank factorizations normal factorizations and singular value decompositions svd there are other factorizations of particular matrices that are natural to certain problems for instance in statistics answers to questions like when is b a a generalized inverse of ab may thus be a computational tool besides they are of significant interest because they provide us with insights into the g inversion of matrix products

moore penrose invertible normal and hermitian elements in rings in this paper we present several new characterizations of normal and hermitian elements in rings with involution in purely algebraic terms and considerably simplify proofs of already existing characterizations

recovery of nonrigid motion and structure the authors introduce a physically correct model of elastic nonrigid motion this model is based on the finite element method but decouples the degrees of freedom by breaking down object motion into rigid and nonrigid vibration or deformation modes the result is an accurate representation for both rigid and nonrigid motion that has greatly reduced dimensionality capturing the intuition that nonrigid motion is normally coherent and not chaotic because of the small number of parameters involved this representation is used to obtain accurate overstrained estimates of both rigid and nonrigid global motion it is also shown that these estimates can be integrated over time by use of an extended kalman filter resulting in stable and accurate estimates of both three dimensional shape and three dimensional velocity the formulation is then extended to include constrained nonrigid motion examples of tracking single nonrigid objects and multiple constrained objects are presented

reverse order law for the moore penrose inverse abstract in this paper we present new results related to the reverse order law for the moore penrose inverse of operators on hilbert spaces some finite dimensional results are extended to infinite dimensional settings

reverse order laws for generalized inverses of multiple matrix products abstract in this article we study reverse order laws for generalized inverses and reflexive generalized inverses of the products of multiple matrices a 1 a n and the products of generalized inverses and reflexive generalized inverses of a n a 1 by applying the multiple product singular value decomposition we obtain necessary and sufficient conditions for one side inclusion relation of reverse order law for generalized inverses and necessary and sufficient conditions of reverse order law for reflexive generalized inverses

bayesian object localisation in images a bayesian approach to intensity based object localisation is presented that employs a learned probabilistic model of image filter bank output applied via monte carlo methods to escape the inefficiency of exhaustive search an adequate probabilistic account of image data requires intensities both in the foreground i e over the object and in the background to be modelled some previous approaches to object localisation by monte carlo methods have used models which we claim do not fully address the issue of the statistical independence of image intensities it is addressed here by applying to each image a bank of filters whose outputs are approximately statistically independent distributions of the responses of individual filters over foreground and background are learned from training data these distributions are then used to define a joint distribution for the output of the filter bank conditioned on object configuration and this serves as an observation likelihood for use in probabilistic inference about localisation the effectiveness of probabilistic object localisation in image clutter using bayesian localisation is illustrated because it is a monte carlo method it produces not simply a single estimate of object configuration but an entire sample from the posterior distribution for the configuration this makes sequential inference of configuration possible two examples are illustrated here coarse to fine scale inference and propagation of configuration estimates over time in image sequences

parametric forms for generalized inverses of markovian kernels and their applications parametric forms for multicondition generalized inverses of i p where p is the transition matrix of a finite irreducible discrete time markov chain are derived these are utilized in techniques for obtaining moments of first passage time distributions

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

recognition by linear combinations of models an approach to visual object recognition in which a 3d object is represented by the linear combination of 2d images of the object is proposed it is shown that for objects with sharp edges as well as with smooth bounding contours the set of possible images of a given object is embedded in a linear space spanned by a small number of views for objects with sharp edges the linear combination representation is exact for objects with smooth boundaries it is an approximation that often holds over a wide range of viewing angles rigid transformations with or without scaling can be distinguished from more general linear transformations of the object by testing certain constraints placed on the coefficients of the linear combinations three alternative methods of determining the transformation that matches a model to a given image are proposed

on the p q outer generalized inverse in banach algebra abstract in this note the p q outer generalized inverse in banach algebra is considered a new representation based on the 1 5 inverse is presented also we construct an iterative method for computing the p q outer generalized inverse in banach algebra

mixing times with applications to perturbed markov chains abstract a measure of the mixing time or time to stationarity in a finite irreducible discrete time markov chain is considered the statistic i j 1 m m ij j where j is the stationary distribution and mij is the mean first passage time from state i to state j of the markov chain is shown to be independent of the initial state i so that i for all i is minimal in the case of a periodic chain yet can be arbitrarily large in a variety of situations an application considering the effects perturbations of the transition probabilities have on the stationary distributions of markov chains leads to a new bound involving for the 1 norm of the difference between the stationary probability vectors of the original and the perturbed chain when is large the stationary distribution of the markov chain is very sensitive to perturbations of the transition probabilities

reverse order laws for the generalized inverses of multiple matrix products by using the ranks of matrices this article gives necessary and sufficient conditions for the reverse product a n a 2a 1of the moore penrose inverses of a1 a2 an to a 1 inverse inner inverse 1 2 inverse reflexive inner inverse 1 3 inverse 1 4 inverse 1 2 3 inverse 1 2 4 inverse or moore penrose inverse of matrix product a a1a2 an

ep elements in rings in this paper we present a number of new characterizations of ep elements in rings with involution in purely algebraic terms and considerably simplify proofs of already existing characterizations

mixed type reverse order law for products of three operators abstract we present new results related to the mixed type reverse order law for the moore penrose inverse of various products of three operators on hilbert spaces some finite dimensional results are extended to infinite dimensional settings

effect of function perturbation on the steady state distribution of genetic regulatory networks optimal structural intervention the dynamics of a rule based gene regulatory network are determined by the regulatory functions in conjunction with whatever probability distributions are involved in network transitions in the case of boolean networks bns and more generally probabilistic boolean networks pbns there has been a significant amount of investigation into the effect of perturbing gene states in particular the design of intervention strategies based on finite or infinite horizon control polices this paper considers the less investigated issue of function perturbations a single function perturbation affects network dynamics and alters the long run distribution whereas any individual gene perturbation has only transient effects and does not change the long run distribution we derive analytic results for changes in the steady state distributions of pbns resulting from modifications to the underlying regulatory rules and apply the derived results to find optimal structural interventions to avoid undesirable states the results are applied to a wnt5a network and a mammalian cell cycle related network respectively to achieve more favorable steady state distributions and reduce the risk of getting into aberrant phenotypes

groups quasidiagonality and interpolation by multiplier projections we relate the following conditions on a a unital c algebra a with the fs property a k1 a 0 b every projection in m a a lifts c the general weyl von neumann theorem holds in m a any selfadjoint element h in m a can be written as h a lpi a for some selfadjoint element a in a some bounded real sequence ai and some mutually orthogonal projections pi in a with ez1 pi 1 d m a has fs and e interpolation by multiplier projections holds for any closed projections p and q in a with pq o there is a projection r in m a such that p b 4 a and that a x b if in addition a is stable combining the above results we obtain counterexamples to the conjecture of g k pedersen a has fs m a has fs for example the stabilized buncededdens algebras hence the generalized weyl von neumann theorem does not generally hold in l ha for a unital c algebras with fs

learning to track the visual motion of contours abstract a development of a method for tracking visual contours is described given an untrained tracker a training motion of an object can be observed over some extended time and stored as an image sequence the image sequence is used to learn parameters in a stochastic differential equation model these are used in turn to build a tracker whose predictor imitates the motion in the training set tests show that the resulting trackers can be markedly tuned to desired curve shapes and classes of motions

computation of moore penrose generalized inverses of matrices with meromorphic function entries abstract in this paper given a field with an involutory automorphism we introduce the notion of moore penrose field by requiring that all matrices over the field have moore penrose inverse we prove that only characteristic zero fields can be moore penrose and that the field of rational functions over a moore penrose field is also moore penrose in addition for a matrix with rational functions entries with coefficients in a field k we find sufficient conditions for the elements in k to ensure that the specialization of the moore penrose inverse is the moore penrose inverse of the specialization of the matrix as a consequence we provide a symbolic algorithm that given a matrix whose entries are rational expression over c of finitely many meromorphic functions being invariant by the involutory automorphism computes its moore penrose inverve by replacing the functions by new variables and hence reducing the problem to the case of matrices with complex rational function entries

reverse order law in c algebras we study equivalent conditions for the reverse order law a1a2 an an a1 a1a2 anan a1 in c algebras as corollaries we obtain some recent and special results

the computation of key properties of markov chains via perturbations abstract computational procedures for the stationary probability distribution the group inverse of the markovian kernel and the mean first passage times of a finite irreducible markov chain are developed using perturbations the derivation of these expressions involves the solution of systems of linear equations and structurally inevitably the inverses of matrices by using a perturbation technique starting from a simple base where no such derivations are formally required we update a sequence of matrices formed by linking the solution procedures via generalised matrix inverses and utilising matrix and vector multiplications four different algorithms are given some modifications are discussed and numerical comparisons are made using a test example the derivations are based upon the ideas outlined by hunter 14

identities concerning the reverse order law for the moore penrose inverse we prove some identities related to the reverse order law for the moore penrose inverse of operators on hilbert spaces extending some results from y tian s cheng linear multilinear algebra 52 2004 and r e cline siam rev vol 6 no 1 1964 to infinite dimensional settings

the moore penrose inverses of products and differences of projections in a c algebra abstract for two given projections p and q in a c algebra we investigate how to express moore penrose inverses of products pq and differences p q and pq qp moreover it is shown that pq qp is moore penrose invertible if and only if pq and p q are moore penrose invertible in addition some related topics are considered

characterizations of generalized inverses associated with markovian kernels abstract a systematic investigation of the various multicondition generalized inverses of i p where p is the transition matrix of a finite irreducible discrete time markov chain is presented the characterizations are based upon the result that i p tu is nonsingular if and only if t and u are chosen so that t 0 where is the stationary probability vector of p and u e 0 where e is the vector of ones partitioned forms for the g inverses are also presented based on a full rank factorization of i p special well known cases such as the group inverses and the moore penrose inverse are given

variances of first passage times in a markov chain with applications to mixing times jm m 1 as a measure of the mixing time or time to stationarity in a finite irreducible discrete time markov chain with stationary distribution pj and mij as the mean first passage time from state i to state j of the markov chain this was shown to be independent of the initial state i with i for all i minimal in the case of a periodic chain yet can be arbitrarily large in a variety of situations in this paper we explore the variance of the mixing time v i starting in state i the v i are shown to depend on i and an exploration of recommended starting states given knowledge of the transition probabilities is considered as a preamble a study of the computation of second moments of the mixing times m ij 2 and the variance of the first passage times in a discrete time markov chain is carried out leading to some new results