# Showing papers in "Linear Algebra and its Applications in 1983"

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TL;DR: In this article, the concepts of strong detectability and strong observability for continuous-time systems are studied and conditions for the existence of observers that estimate unmeasured outputs on the basis of partial information on the input and the state.

Abstract: Conditions are given for the existence of observers that estimate unmeasured outputs on the basis of partial information on the input and the state. The concepts of strong detectability and strong observability, introduced before in the literature for discrete-time systems only, are defined and studied for continuous-time systems. It is shown that there are two different concepts of strong detectability, which coincide for discrete-time systems. Algebraic conditions for either concept are given. It is shown that these concepts are intimately related to the existence of strong observers, i.e. observers that only use the output of the plant.

321 citations

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TL;DR: This paper takes a unified approach to the partial realization problem in which it seeks to incorporate ideas from numerical linear algebra, most of which were originally developed in other contexts.

Abstract: In this paper we take a unified approach to the partial realization problem in which we seek to incorporate ideas from numerical linear algebra, most of which were originally developed in other contexts. We approach the partial realization problem from several different angles and explore the connections to such topics as factoriza- tion of Hankel matrices, block tridiagonalization, generalizations of the Lanczos process for biorthogonalization, the Euclidean algorithm and the principal-part con- tinued fractions of Arne Magnus, the Pade table, and the Berlekamp-Masseyalgo- rithm. In this way we are able to clarify some previous results by Rissanen, Kalman, and others and place them in a broader context. This leads to several results and concepts which we think are new. Our analysis is restricted to the scalar case, but some definitions and formulations have been rigged to facilitate an extension to the matrix case.

291 citations

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TL;DR: The typical rank (= maximal border rank) of tensors of a given size and the set of optimal bilinear computations of typical TensorRank are investigated and it is shown that for the size ( n, n, 3) with n odd, the complement of theSet of Tensors of maximal borderRank is a hypersurface.

Abstract: The typical rank (= maximal border rank) of tensors of a given size and the set of optimal bilinear computations of typical tensors of a given rank are investigated. For the size ( n , n , 3) with n odd, the complement of the set of tensors of maximal border rank is a hypersurface. Its equation is given.

245 citations

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TL;DR: In this paper, the general continuous-time linear-quadratic control problem is considered, and it is shown that recently developed linear system theoretic properties and algorithms play an important role in solving this singular control problem.

Abstract: The general continuous-time linear-quadratic control problem is considered. It is shown that recently developed linear system theoretic properties and algorithms play an important role in solving this singular control problem.

216 citations

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TL;DR: Two algorithms, based on the bisection technique and Newton's method, are shown to be very fast for computing the eigenvalues of a 7− or 5-diagonal BST-matrix.

Abstract: We are investigating spectral properties of band symmetric Toeplitz matrices (BST matrices). By giving a suitable representation of a BST matrix, we achieve separation results and multiplicity conditions for the eigenvalues of a 7°r 5-diagonal BST matrix and also structural properties of the eigenvectors. We give eigenvalue bounds for a 2k 1-diagonal BST matrix and also necessary and sufficient conditions for positive definiteness which are easy to check. The same conditions apply in the case of block BST matrices, either with full blocks or with BST blocks. We exhibit fast computational methods for the evaluation of the determinant and the characteristic polynomial of a BST matrix, either for sequential or for parallel computations. Two algorithms, based on the bisection technique and Newton's method, are shown to be very fast for computing the eigenvalues of a 7− or 5-diagonal BST-matrix.

157 citations

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TL;DR: The method is based on the Schur factorization A = QSQ H and uses a fast recursion to compute the upper triangular square root of S and it is shown that if α = ∥ X ∥ 2 /∥ A ∥ is not large, then the computed square root is the exact square root

Abstract: A fast and stable method for computing the square root X of a given matrix A ( X 2 = A ) is developed. The method is based on the Schur factorization A = QSQ H and uses a fast recursion to compute the upper triangular square root of S . It is shown that if α = ∥ X ∥ 2 /∥ A ∥ is not large, then the computed square root is the exact square root of a matrix close to A . The method is extended for computing the cube root of A . Matrices exist for which the square root computed by the Schur method is ill conditioned, but which nonetheless have well-conditioned square roots. An optimization approach is suggested for computing the well-conditioned square roots in these cases.

150 citations

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TL;DR: In this article, it was shown that the spectrum of B can be obtained from that of A (multiplicities counted) by moving each eigenvalue by at most ϵ.

Abstract: Let A and B be normal operators on a Hilbert space Let K A and K B be subsets of the complex plane, at distance at least δ from each other; let E be the spectral projector for A belonging to K A , and let F be the spectral projector for B belonging to K B Our main results are estimates of the form δ ‖ EF ‖ c ‖ E ( A − B ) F ‖; in some special situations, the constant c is as low as 1 As an application, we prove, for an absolute constant d , that if the space is finite-dimensional and if A and B are normal with ‖;A − B‖ ⩽ ϵ d , then the spectrum of B can be obtained from that of A (multiplicities counted) by moving each eigenvalue by at most ϵ Our main results have equivalent formulations as statements about the operator equation AQ - QB = S Let A and B be normal operators on perhaps different Hilbert spaces Assume σ ( A ) K A and σ ( B ) K B , where K A , K B , and δ are as before Then we give estimates of the forms δ ‖ Q ‖⩽ c ‖ AQ − QB ‖

144 citations

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TL;DR: In this article, the zero structure, the polar structure, and the left and right null space structure of a polynomial matrix P(λ) have been computed using a new numerical method.

Abstract: We give a new numerical method to compute the eigenstructure—i.e. the zero structure, the polar structure, and the left and right null space structure—of a polynomial matrix P(λ). These structural elements are of fundamental importance in several systems and control problems involving polynomial matrices. The approach is more general than previous numerical methods because it can be applied to an arbitrary m × n polynomial matrix P(λ) with normal rank r smaller than m and/or n. The algorithm is then shown to compute the structure of the left and right null spaces of P(λ) as well. The speed and accuracy of this new approach are also discussed.

137 citations

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TL;DR: In this article, the authors give a formal treatment of determinantal identities of the minors of a matrix and give a common, concise derivation of some important determinantial identities attributed to the mathematicians in the title.

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.

132 citations

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TL;DR: In this paper, the authors study spaces generated by translations of a fixed function over lattice points and provide algebraic and approximation properties for these spaces which show their applicability for finite element analysis.

Abstract: We study spaces generated by translations of a fixed function over lattice points. We provide some new algebraic and approximation properties for these spaces which show their applicability for finite element analysis.

131 citations

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TL;DR: Various representations for systems described by aset of high-order differential equations of the form R0w + R1w + … + Rsw(s) = 0, with R0, R1,…, Rs not necessarily square matrices are developed.

Abstract: In this paper we develop various representations for systems described by aset of high-order differential equations of the form R0w + R1w + … + Rsw(s) = 0, with R0, R1,…, Rs not necessarily square matrices. The variables w are the external variables. Particular attention is paid to the problem of obtaining minimal state-space realizations and input-output or input-state-output representations of such systems.

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TL;DR: A unifying framework is revealed where several known results fit naturally and special attention is given to the embedding problem of the Lyapunov equation in view of its direct application to generalized Levinson algorithms.

Abstract: This paper is concerned with a systematic approach to the properties of ∑-lossless rational transfer functions in the discrete as well as in the continuous time case. As a result, a unifying framework is revealed where several known results fit naturally. Special attention is given to the embedding problem of the Lyapunov equation in view of its direct application to generalized Levinson algorithms.

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TL;DR: In this article, a criterion for copositive matrices is given and for n = 3 the set of all copositives is determined in terms of matrix elements, and the problem of excluding periodic solutions of certain algebraic differential equations is considered.

Abstract: A criterion for copositive matrices is given and for n = 3 the set of all copositive matrices is determined in terms of matrix elements. Copositive matrices are applied to the problem of excluding periodic solutions of certain algebraic differential equations.

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TL;DR: In this article, a detailed study of symmetric transfer functions is presented, and a detailed analysis of transfer functions can be found in Section 5.1.1]...

Abstract: A detailed study of symmetric transfer functions is presented.

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TL;DR: In this article, it was shown that the generalized conjugate-gradient (CC) can be simplified if a nonsingular matrix H is available such that HA = ATH.

Abstract: The conjugate-gradient (CC) method, developed by Hestenes and Stiefel in 1952, can be effectively used to solve the linear system Au = b when A is symmetrixable in the sense that ZA and Z are symmetric and positive definite (SPD) for some Z. A number of generalizations of the CG method have been proposed by the authors and by others for handling the nonsymmetrizable case. For many problems the amount of computer memory and computational effort required may be so large as to make the procedures not feasible. Truncated schemes are often used, but in some cases the truncated methods may not converge even though the nontruncated schemes converge. However, it is well known that if A is symmetric, the generalized CG schemes can be greatly simplified, even though A is not SPD, so that the truncated schemes are equivalent to the nontruncated schemes. In the present paper it is shown that such a simplification can occur if a nonsingular matrix H is available such that HA = ATH. (Of course, if A = AT, then H can be taken to be the identity matrix.) It is also shown that such an H always exists; however, it may not be practical to compute H. These results are used to derive three variations of the Lanczos method for solving nonsymmetrizable systems. Two of the forms are well known, but the third appears to be new. An argument is given for choosing the third form over the other two.

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TL;DR: In this paper, a class of special, highly efficient multigrid methods for solving h-discrete elliptic differential equations is introduced, which are characterized by intermediate grids (between the given h-grid and the 2h-grid) and by special fine-to-coarse and coarse-tofine grid transfer techniques.

Abstract: By MGR we denote a class of special, highly efficient multigrid methods for solving h-discrete elliptic differential equations. Unlike standard multigrid methods [4, 9, 18], MGR methods are characterized by “intermediate” grids (between the given h-grid and the 2h-grid) and by special fine-to-coarse and coarse-to-fine grid transfer techniques. The MGR idea has been conceived [13, 7] by the second author in trying to extend the range of applicability of the total-reduction method [15, 16] to more general problems. Described in a somewhat different way, methods of MGR type have in the meantime also been considered by Braess [1-3] and Meis [11]. The convergence properties of the simplest MGR method (MGR-0) are essentially improved if it is combined with one step of checkered Gauss-Seidel iteration (MGR-CH). For the model problem of Poisson's equation in the unit square, for example, the spectral radius decreases from 12 to 227 by such a modification, whereas the computational effort is not essentially enlarged. For Poisson-like equations, MGR-CH yields the fastest iterative solver known so far. Other MGR variants are particularly suitable for anisotropic operators.

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TL;DR: In this article, the condition number of the n X n matrix p = [Pi-ltxj)lY,Y,j-l is investigated, where p,((.) = p, (.; dX) are orthogonal polynomials with respect to some weight distribution dX, and xi are pairwise distinct real numbers.

Abstract: The condition number (relative to the Frobenius norm) of the n X n matrix p” = [Pi-ltxj)lY,j-l is investigated, where p,( .) = p,( .; dX) are orthogonal polynomials with respect to some weight distribution dX, and xi are pairwise distinct real numbers. If the nodes x j are the zeros of p,, , the condition number is either expressed, or estimated from below and above, in terms of the Christoffel numbers for dh, depending on whether the p, are normalized or not. For arbitrary real xi and normalized p, a lower bound of the condition number is obtained in terms of the Christoffel function evaluated at the nodes. Numerical results are given for minimizing the condition number as a function of the nodes for selected classical distributions dX.

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TL;DR: Several variants of Gram-Schmidt orthogonalization are reviewed from a numerical point of view in this paper, and it is shown that the classical and modified variants correspond to the Gauss-Jacobi and Gauss -Seidel iterations for linear systems.

Abstract: Several variants of Gram-Schmidt orthogonalization are reviewed from a numerical point of view. It is shown that the classical and modified variants correspond to the Gauss-Jacobi and Gauss-Seidel iterations for linear systems. Further it is shown that orthogonalization with respect to elliptic norms and biorthogonalization can be formulated as orthogonalization by oblique projections.

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TL;DR: This paper presents two methods not requiring the explicit knowledge of the roots of r and obtains various properties of the similarity transformations between Jacobi matrices, which are proved by simple matrix calculus without using the generalized Christoffel theorem.

Abstract: Given a Jacobi matrix, the problem in question is to find the Jacobi matrix corresponding to the weight function modified by a polynomial r . Galant and Gautschi derived algorithms, based on the generalized Christoffel theorem of Uvarov, applicable when the roots of r are known. In this paper we present two methods not requiring the explicit knowledge of the roots of r . We also obtain various properties of the similarity transformations between Jacobi matrices, which we prove by simple matrix calculus without using the generalized Christoffel theorem.

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TL;DR: The class of multivariate normal densities n (0, Σ ) whose inverse covariance matrix Σ −1 is an M-matrix is equivalent to this normal density being multivariate totally positive of order 2(MTP 2 ) as mentioned in this paper.

Abstract: The class of multivariate normal densities n (0, Σ ) whose inverse covariance matrix Σ ) −1 is an M-matrix is equivalent to this normal density being multivariate totally positive of order 2(MTP 2 ). Equivalent characterizations are given in terms of certain partial correlation coefficients being positive. It is further shown that related partial and multiple regression coefficients and canonical correlation are positive. When Σ is an M -matrix the corresponding normal random vector components are negatively associated . This concept and some extensions are discussed.

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TL;DR: In this article, it was shown that such a B exists with m⩽ 1 2 k(k+1)−N, where 2N is the maximal number of entries which equal zero in a nonsingular principal submatrix of A.

Abstract: Let A be a real symmetric n × n matrix of rank k, and suppose that A = BB′ for some real n × m matrix B with nonnegative entries (for some m). (Such an A is called completely positive.) It is shown that such a B exists with m⩽ 1 2 k(k+1)−N , where 2N is the maximal number of (off-diagonal) entries which equal zero in a nonsingular principal submatrix of A. An example is given where the least m which works is (k+1) 2 4 (k odd), k(k+2) 4 (k even).

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TL;DR: In this article, it was shown that a matrix A admits a 1-inverse if and only if a linear combination of all the r × r minors of A is equal to one, where r is the rank of A.

Abstract: It is proved that a matrix A over an integral domain admits a 1-inverse if and only if a linear combination of all the r × r minors of A is equal to one, where r is the rank of A. Some results on the existence of Moore-Penrose inverses are also obtained.

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TL;DR: In this paper, a complete description including multiplicity is given for the Jordan structure of a matrix which is a small perturbation of a known Jordan structure, and the problem solved here was solved independently and the other solution has been published in English.

Abstract: In this paper a complete description including multiplicity is given for the Jordan structure of a matrix which is a small perturbation of a matrix of known Jordan structure. The problem solved here was solved independently, as noted, and the other solution has been published in English. However, the solutions are different, the problem is important, and it seems desirable to have this solution widely available.

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TL;DR: In this article, the iterative aggregation method for the solution of a system of linear algebraic equations x = Ax + b, where A ≥ 0, b ≥ 0 and s > 0, and s < s, is proved to be locally convergent.

Abstract: The iterative aggregation method for the solution of a system of linear algebraic equations x = Ax + b, where A ≥ 0, b ≥ 0, s > 0, and s ′ A < s ′, is proved to be locally convergent. It is shown that the method can be considered a consistent nonstationary iterative method, where the iteration matrix depends on the current iterate, and that some norm of the iteration matrix is less than one in the vicinity of the solution.

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TL;DR: A robust Lanczos algorithm is presented which is fast, easy to understand, uses about 30 words of extra storage, and has a fairly short program.

Abstract: The Lanczos algorithm is used to compute some eigenvalues of a given symmetric matrix of large order. At each step of the Lanczos algorithm it is valuable to know which eigenvalues of the associated tridiagonal matrix have stabilized at eigenvalues of the given symmetric matrix. We present a robust algorithm which is fast (20 j to 40 j operations at the j th Lanczos step), uses about 30 words of extra storage, and has a fairly short program (approxiamately 200 executable statements)

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TL;DR: In this paper, the sign matrices uniquely associated with the matrices (M − ζ j I ) 2, where the corners of a rectangle oriented at π /4 to the axes of a Cartesian coordinate system, were used to compute the number of eigenvalues of the arbitrarily chosen matrix M which lie within the rectangle, and to determine the left and right invariant subspaces of M associated with these eigen values.

Abstract: The sign matrices uniquely associated with the matrices ( M − ζ j I ) 2 , where ζ j are the corners of a rectangle oriented at π /4 to the axes of a Cartesian coordinate system, may be used to compute the number of eigenvalues of the arbitrarily chosen matrix M which lie within the rectangle, and to determine the left and right invariant subspaces of M associated with these eigenvalues. This paper is concerned with the proof of this statement, and with the details of the computation of the required sign matrices.

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TL;DR: In this paper, a new algorithm for the numerical calculation of the singular values and vectors of an infinite Hankel matrix of known finite rank r is presented, which proceeds by reduction to the singular value problem for an r × r matrix, achieved without solving for the poles of the symbol of H.

Abstract: Let H be an infinite Hankel matrix of known finite rank r . A new algorithm for the numerical calculation of the singular values and vectors of H is presented. The method proceeds by reduction to the singular value problem for an r × r matrix; this is achieved without solving for the poles of the symbol of H . The resulting algorithm is of order r 3 .

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TL;DR: In this article, the Euler-MacLaurin summation formula is used to deduce the Whittaker-Shannon sampling theorem for not necessarily band-limited functions, as well as to study numerical integration over the real axis.

Abstract: The Euler-MacLaurin summation formula is used to deduce the Whittaker-Shannon sampling theorem for not necessarily band-limited functions, as well as to study numerical integration over the real axis. Concerning the latter, error estimates are determined in case the function to be integrated is smooth but not necessarily analytic. Two characteristic examples are given.

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TL;DR: In this article, the authors present an algorithm polvol for the computation of the d-dimensional volume V(P) of a bounded polyhedron P ⊂ R d, which is determined by the local properties of P at all its vertices.

Abstract: We present an algorithm polvol for the computation of the d-dimensional volume V(P) of a bounded polyhedron P ⊂ R d. It is shown that V(P) is determined by the local properties of P at all its vertices. So our method consists in letting a hyperplane “sweep” through R d, collecting the local information available at every vertex ak of P. This leads to an additive contribution to the volume from every ak, the sum of these contributions being V(P). It is assumed that P is represented in Boolean form as described in [13].

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TL;DR: In this paper, a linear combination of n -by-n Hermitian matrices over the complex field was shown to be possible in a finite number of steps, where A and B are real symmetric sparse matrices.

Abstract: Let A and B be n -by- n Hermitian matrices over the complex field. A result of Au-Yeung [1] and Stewart [8] states that if x ∗ (A + iB)x≠0 for all nonzero n -vectors x , then there is a linear combination of A and B which is positive definite. In this article we present an algorithm which finds such a linear combination in a finite number of steps. We also discuss the implementation of the algorithm in case A and B are real symmetric sparse matrices.