sufficient or necessary conditions for modal controllability and observability of roesser s 2 d system model controllability and observability properties of roesser s 2 d system model have been investigated and some strong necessary and or sufficient conditions are found

corner cutting algorithms for the bezier representation of free form curves abstract we give a general treatment of the problem of converting a curve expressed in a b spline expansion into its bezier form we show that corner cutting of the b spline control polygon leads to the bezier points exactly when the bezier matrix is totally positive also for spline spaces used to model space curves with curvature continuity we give an algorithm for determining the bezier representation

an optimal test for almost strict total positivity abstract a matrix is almost strictly totally positive if all its minors are nonnegative and they are positive if and only if they do not contain a zero in their diagonal an optimal test to check if a given matrix belongs to this class of matrices is presented for this purpose we establish a bijection between the set of nonzero entries of the matrix and a set of submatrices called essential submatrices which are explicitly constructed the test shows that it is sufficient to check the positivity of the essential minors improving the characterization presented in 15 essential minors are also applied to the construction of accurate bidiagonal decompositions of almost strictly totally positive matrices which in turn can be used for deriving accurate algorithms for these matrices

on the cayley hamilton theorem for two dimensional systems an alternate proof of the 2 d cayley hamilton theorem is supplied together with an extension to derived polynomials

qr decomposition of almost strictly sign regular matrices in this paper l d u factorization of nonsingular sign regular and almost strictly sign regular matrices is analyzed besides nonsingular almost strictly sign regular matrices are characterized through the q r factorization illustrative numerical examples are included

state space discrete systems 2 d eigenvalues a correction is given to a proof of the fact presented in a paper by roesser 1 that every partitioned matrix a not only satisfies the two dimensional characteristic equation but it must also satisfy an additional set of equations a new definition of 2 d eigenvalues is proposed

structure of invertible bi infinite totally positive matrices an invertible nonfinite totally positive matrix a is shown to have one and only one main diagonal this means that exactly one diagonal of a has the property that all finite sections of a principal with respect to this diagonal are invertible and their inverses converge boundedly and entrywise to a i this is shown to imply restrictions on the possible shapes of such a matrix in the proof such a matrix is also shown to have an la invertible ldu factorization in addition decay of the entries of such a matrix away from the main diagonal is demonstrated it is also shown that a bounded sign regular matrix carrying some bounded sequence to a uniformly alternating sequence must have all its columns in co

using the cayley hamilton theorem with n partitioned matrices a definition is given for the characteristic equation of an n partitioned matrix it is then proved that this matrix satisfies its own characteristic equation this can then be regarded as a version of the cayley hamilton theorem of use with n dimensional systems

generalization of cayley hamilton theorem for multivariate rational matrices the two equivalent generalizations of the famous cayley hamilton theorem for multivariate rational matrices over any number field are proposed which are also generalizations of the nine known generalized cayley hamilton theorems restricting our results to a smaller range leads to the two equivalent generalized cayley hamilton theorems for multivariate polynomial matrices over any number field one of which is more efficient in some cases than a known result

almost strictly totally negative matrices a real matrix a a i j 1 i j n is said to be almost strictly totally negative if it is almost strictly sign regular with signature e 1 1 1 which is equivalent to the property that all its nontrivial minors are negative in this paper an algorithmic characterization of nonsingular almost strictly totally negative matrices is presented

computation of the transfer function matrix and its laurent expansion of generalized two dimensional systems an algorithm is developed for the computation of the transfer function matrix of a two dimensional system which is given in its generalized form the algorithm is a recursion in terms of the original system matrix and does not require the inversion of a two variable polynomial matrix an algorithm for the evaluation of the laurent expansion of the inverse of a two variable polynomial matrix is also presented

hadamard powers and totally positive matrices considered are continuous positive hadamard powers of entry wise positive nonnegative matrices those that are eventually in the sense of all hadamard powers beyond some point totally positive totally nonnegative doubly nonnegative and doubly positive are characterized for example for matrices with at least four rows and four columns hadamard powers greater than one of totally positive matrices need not be totally positive but they are eventually totally positive

factorization of totally positive symmetric periodic banded matrices with applications abstract we show that a totally positive symmetric periodic banded matrix a can be factored in a symmetric manner into positive one banded periodic factors and deduce from this that a has positive eigenvalues with certain restrictions on their coincidence and on the sign changes of the corresponding eigenvectors from this are derived spectral properties of certain positive periodic spline operators

factorization of banded lower triangular infinite matrices abstract it is shown that a lower triangular strictly m banded infinite matrix can be factored into a product of m lower triangular strictly 1 banded matrices a method to construct such a factorization is described

the inverse of a totally positive biinfinite band matrix it is shown that a bounded bi infinite banded totally positive matrix a is boundedly invertible iff there is one and only one bounded sequence mapped by a to the sequence the argument shows that such a matrix has a main diagonal i e the inverse of a is the bounded pointwise limit of inverses of finite sections of a principal with respect to a particular diagonal hence 4 j 7 or its inverse is again totally positive 0 introduction this paper is a further step in a continuing effort to understand certain linear spline approximation schemes convergence of such processes is intimately tied to their stability i e to their boundedness as maps on c say use of the b spline basis shows this question to be equivalent to bounding the inverse of certain totally positive band matrices the calculation of bounds on the inverse of a given matrix is in general a difficult task it is hoped that the present investigation into the consequences of bandedness and total positivity for the structure of the inverse may ultimately prove helpful in obtaining such bounds the results in this paper were obtained in the study of a conjecture due to c a micchelli 7 in connection with his work on the specific approximation scheme of interpolation at a strictly increasing point sequence t by elements of sm t i e by splines of some order m with some knot sequence t micchelli became convinced that every bounded function has one and only one bounded spline interpolant iff the particular function which satisfies t all has a bounded spline interpolant in sm if ni ni mt denotes the corresponding b spline basis for sm then micchelli s conjecture can be phrased thus the matrix

structural invariants of two dimensional systems in this paper some fundamental structural properties of two dimensional 2 d systems which remain invariant under feedback and output injection transformation groups are identified and investigated for the first time as is well known structural invariants that follow from the definition of controlled and conditioned invariance output nulling input containing self bounded and self hidden subspaces play pivotal roles in many theoretical studies of systems theory and in the solution of several control estimation problems these concepts are developed and studied within a 2 d context in this paper

the approximation of a totally positive band matrix by a strictly banded totally positive one every nonsingular totally positive m banded matrix is shown to be the product of m totally positive one banded matrices and therefore the limit of strictly m banded totally positive matrices this result is then extended to bi infinite m banded totally positive matrices with linearly independent rows and columns in the process such matrices are shown to possess at least one diagonal whose principal sections are all nonzero as a consequence such matrices are seen to be approximable by strictly m banded totally positive ones

determinantal criteria for total positivity a matrix is totally positive respectively strictly totally positive if all its minors are nonnegative respectively positive in this paper we revisit the main determinantal criteria for these matrices and provide two new ones one of them for the total positivity of m n matrices with any n consecutive rows linearly independent and another one to check if an n n totally positive matrix has all minors of order less than k k n positive

shape preserving properties of ω q bernstein polynomials abstract in this paper we discuss shape preserving properties of the q bernstein polynomials b n q f x introduced by lewanowicz and wozny in s lewanowicz p wo ny generalized bernstein polynomials bit 44 1 2004 63 78 for q 0 1 when 0 we recover the q bernstein polynomials introduced by phillips g m phillips bernstein polynomials based on the q integers ann numer math 4 1997 511 518 when q 1 we recover the classical bernstein polynomials for q 0 1 we show that the basic q bernstein polynomial basis is a normalized totally positive basis on 0 1 and that the q bernstein operators b n q on c 0 1 are variation diminishing monotonicity preserving and convexity preserving we also show that the q bernstein polynomials of a convex function f in the case q 0 1 are monotonic in the parameters and q for fixed n

intervals of almost totally positive matrices we consider the class of the totally nonnegative matrices i e the matrices having all their minors nonnegative and intervals of matrices with respect to the chequerboard partial ordering which results from the usual entrywise partial ordering if we reverse the inequality sign in all components having odd index sum for these intervals in 1982 we stated in this journal the following conjecture if the left and right endpoints of an interval are nonsingular and totally nonnegative then all matrices taken from the interval are nonsingular and totally nonnegative in this paper we show that this conjecture is true if we restrict ourselves to the subset of the almost totally positive matrices

q bernstein polynomials and bezier curves we define q bernstein polynomials which generalize the classical bernstein polynomials and show that the difference of two consecutive q bernstein polynomials of a function f can be expressed in terms of second order divided differences of f it is also shown that the approximation to a convex function by its q bernstein polynomials is one sided a parametric curve is represented using a generalized bernstein basis and the concept of total positivity is applied to investigate the shape properties of the curve we study the nature of degree elevation and degree reduction for this basis and show that degree elevation is variation diminishing as for the classical bernstein basis

a factorization theorem for banded matrices in this paper we prove a factorization theorem for strictly m banded totally positive matrices we show that such a matrix is a product of m one banded matrices with positive entries a bi infinite matrix a ai i ca i and ai i n ai i o these matrices arise quite freqiently in applications and recently the have been the subject of several independent investigations 2 3 in this paper we prove a factorization theorem for strictly m handed totally positive matrices we show that such a matrix is a product of m one banded matrices with positive entries let us begin by recalling that a matrix a is totally positive if and only if ull its minors are nonnegative

further applications of the cauchon algorithm to rank determination and bidiagonal factorization abstract for a class of matrices connected with cauchon diagrams cauchon matrices and the cauchon algorithm a method for determining the rank and for checking a set of consecutive row or column vectors for linear independence is presented cauchon diagrams are also linked to the elementary bidiagonal factorization of a matrix and to certain types of rank conditions associated with submatrices called descending rank conditions

m dimensional cayley hamilton theorem the theorem states that every block square matrix satisfies its own m d m dimensional m or 1 matrix characteristic polynomial the exact statement and a simple proof of this theorem are given the theorem refers to a matrix a subdivided into m blocks and hence having dimension at least m the conclusion is that every square matrix a with dimension m satisfies several m d characteristic matrix polynomials with degrees n sub 1 n sub m such that n sub 1 n sub m

determinantal identities gauss schur cauchy sylvester kronecker jacobi binet laplace muir and cayley abstract we give a common concise derivation of some important determinantal identities attributed to the mathematicians in the title we also give a formal treatment of determinantal identities of the minors of a matrix

totally nonnegative 0 1 matrices abstract we investigate 0 1 matrices which are totally nonnegative and therefore which have all of their eigenvalues equal to nonnegative real numbers such matrices are characterized by four forbidden submatrices of orders 2 and 3 we show that the maximum number of 0 s in an irreducible totally nonnegative 0 1 matrix of order n is n 1 2 and characterize those matrices with this number of 0 s we also show that the minimum perron value of an irreducible totally nonnegative 0 1 matrix of order n equals 2 2 cos 2 n 2 and characterize those matrices with this perron value

interpolatory and variation diminishing properties of generalized b splines on considere des classes de fonctions satisfaisant certains criteres simples de signe et de regularite et une propriete de decomposition on montre que certaines fonctions b splines trigonometriques possedent ces proprietes on etudie l interpolation sur un ensemble fini de n uds et sur un ensemble de n uds infini espaces periodiquement