A useful representation of fractional order systems is the state space representation. For the linear fractional systems of commensurate order, the state space representation is defined as for regular integer state sp...
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A useful representation of fractional order systems is the state space representation. For the linear fractional systems of commensurate order, the state space representation is defined as for regular integer state space representation with the state vector differentiated to a real order. This paper presents a solution of the linear fractional order systems of commensurate order in the state space. The solution is obtained using a technique based on functions of square matrices and the Cayley-Hamilton theorem. The technique developed for linear systems of integer order is extended to derive analytical solutions of linear fractional systems of commensurate order. The basic ideas and the derived formulations of the technique are presented. Both, homogeneous and inhomogeneous cases with usual input functions are solved. The solution is calculated in the form of a linear combination of suitable fundamental functions. The presented results are illustrated by analyzing some examples to demonstrate the effectiveness of the presented analytical approach. (C) 2011 Elsevier Ltd. All rights reserved.
We consider restricted rational Lanczos approximations to matrix functions representable by some integral forms. A convergence analysis that stresses the effectiveness of the proposed method is developed. Error estima...
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We consider restricted rational Lanczos approximations to matrix functions representable by some integral forms. A convergence analysis that stresses the effectiveness of the proposed method is developed. Error estimates are derived. Numerical experiments are presented. Copyright (C) 2008 John Wiley & Sons, Ltd.
The paper deals with Krylov methods for approximating functions of matrices via interpolation. In this frame residual smoothing techniques based on quasi-kemel polynomials are considered. Theoretical results as well a...
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The paper deals with Krylov methods for approximating functions of matrices via interpolation. In this frame residual smoothing techniques based on quasi-kemel polynomials are considered. Theoretical results as well as nurnerical experiments illustrate the effectiveness of our approach. Copyright 0 2004 John Wiley & Sons, Ltd.
For the engineering community, Gray's tutorial monograph on Toeplitz and circulant matrices has been, and remains, the best elementary introduction to the Szego theory on large Toeplitz matrices. In this paper, th...
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For the engineering community, Gray's tutorial monograph on Toeplitz and circulant matrices has been, and remains, the best elementary introduction to the Szego theory on large Toeplitz matrices. In this paper, the most important results of the cited monograph are generalized to block Toeplitz (BT) matrices by maintaining the same mathematical tools used by Gray, that is, by using asymptotically equivalent sequences of matrices. As applications of these results, the geometric minimum mean square error (MMSE) for both an infinite-length multi-variate linear predictor and an infinite-length decision feedback equalizer (DFE) for multiple-input-multiple-output (MIMO) channels, are obtained as a limit of the corresponding finite-length cases. Similarly, a short derivation of the well-known capacity of a time-invariant MIMO Gaussian channel with intersymbol interference (ISI) and fixed input covariance matrix is also presented.
Toeplitz matrices and functions of Toeplitz matrices (such as the inverse of a Toeplitz matrix, powers of a Toeplitz matrix or the exponential of a Toeplitz matrix) arise in many different theoretical and applied fiel...
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Toeplitz matrices and functions of Toeplitz matrices (such as the inverse of a Toeplitz matrix, powers of a Toeplitz matrix or the exponential of a Toeplitz matrix) arise in many different theoretical and applied fields. They can be found in the mathematical modeling of problems where some kind of shift invariance occurs in terms of space or time. R. M. Gray's excellent tutorial monograph on Toeplitz and circulant matrices has been, and remains, the best elementary introduction to the Szego distribution theory on the asymptotic behavior of continuous functions of Toeplitz matrices. His asymptotic results, widely used in engineering due to the simplicity of its mathematical proofs, do not concern individual entries of these matrices but rather, they describe an "average" behavior. However, there are important applications where the asymptotic expressions of interest are directly related to the convergence of a single entry of a continuous function of a Toeplitz matrix. Using similar mathematical tools and to gain insight into the solutions of this sort of problems, the present correspondence derives new theoretical results regarding the convergence of these entries.
The matrix exponential is a very important subclass of functions of matrices that has been studied extensively in the last 50 years. In this thesis, we discuss some of the more common matrix functions and their genera...
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The matrix exponential is a very important subclass of functions of matrices that has been studied extensively in the last 50 years. In this thesis, we discuss some of the more common matrix functions and their general properties, and we specifically explore the matrix exponential. In principle, the matrix exponential could be computed in many ways. In practice, some of the methods are preferable to others, but none are completely satisfactory. Computations of the matrix exponential using Taylor Series, Scaling and Squaring, Eigenvectors, and the Schur Decomposition methods are provided.
Having origins in the increasingly popular Matrix Theory, the square root function of a matrix has received notable attention in recent years. In this thesis, we discuss some of the more common matrix functions and th...
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Having origins in the increasingly popular Matrix Theory, the square root function of a matrix has received notable attention in recent years. In this thesis, we discuss some of the more common matrix functions and their general properties, but we specifically explore the square root function of a matrix and the most efficient method (Schur decomposition) of computing it. Calculating the square root of a 2×2 matrix by the Cayley-Hamilton Theorem is highlighted, along with square roots of positive semidefinite matrices and general square roots using the Jordan Canonical Form.
In earlier works, authors such as Varga, Micchelli and Willoughby, Ando, and Fiedler and Schneider have studied and characterized functions which preserve the M-matrices or some subclasses of the M-matrices, such as t...
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In earlier works, authors such as Varga, Micchelli and Willoughby, Ando, and Fiedler and Schneider have studied and characterized functions which preserve the M-matrices or some subclasses of the M-matrices, such as the Stieltjes matrices. Here we characterize functions which either preserve the inverse M-matrices or map the inverse M-matrices to the M-matrices. In one of our results we employ the theory of Pick functions to show that if A and B are inverse M-matrices such that B-1 <= A(-1), then (B + tI)(-1) <= (A + tI)(-1), for all t >= 0.
An analytical function f(A) of an arbitrary n x n constant matrix A is determined and expressed by the "fundamental formula", the linear combination Of Constituent matrices. The constituent matrices Z(kh), w...
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An analytical function f(A) of an arbitrary n x n constant matrix A is determined and expressed by the "fundamental formula", the linear combination Of Constituent matrices. The constituent matrices Z(kh), which depend on A but not on the function f(s), are computed from the given matrix A, that may have repeated eigenvalues. The associated companion matrix C and Jordan matrix J are then expressed when all the eigenvalues with multiplicities are known. Several other related matrices, such as Vandermonde matrix V, modal matrix W, Krylov matrix K and their inverses, are also derived and depicted as in a 2-D or 3-D mapping diagram. The constituent matrices Z(kh) of A are thus obtained by these matrices through similarity matrix transformations. Alternatively, efficient and direct approaches for Z(kh), can be found by the linear combination of matrices, that may be further simplified by writing them in "super column matrix" forms. Finally, a typical example is provided to show the merit of several approaches for the Constituent matrices of a given matrix A. (C) 2003 Elsevier Inc. All rights reserved.
An inequality is proved for convex functions applied to self-adjoint matrices, Several known inequalities are shown to be consequences, but properly weaker. (C) 2001 Elsevier Science Inc. All rights reserved.
An inequality is proved for convex functions applied to self-adjoint matrices, Several known inequalities are shown to be consequences, but properly weaker. (C) 2001 Elsevier Science Inc. All rights reserved.
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