Article

On explicit formulas of the principal matrix pth root by polynomial decompositions

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  • Universidade Federal de Mato Grosso do Sul. Campo Grande - Brazil
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Abstract

We present some explicit formulas for calculating the principal pth root of a square matrix. The main tools are based on various polynomial decompositions of the principal matrix pth root and well-known properties of the linear recursive sequences.

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... For example, this matrix is involved in control and systems theory, matrix differential equations, nonlinear matrix equations, finance and health care. Many methods and techniques have been expanded to provide exact and approximate representations of the matrix pth root (see [1], [4], [9], [17], and references therein). In this study, we consider the Fibonacci-Hörner decomposition of the matrix powers (see [2], [5], [6] and [7]) and some techniques based on some properties of generalized Fibonacci sequences (see [10] and [18]), to provide some explicit formulas of the matrix pth roots. ...
... A matrix pth root can be defined using several definitions of a matrix function of the current literature (see [11], [12], [13], [16] and [19]). In general, a matrix pth root may not exist or there may be an infinite number of solutions for (1). In this study, we are particularly interested in the polynomial solutions of Equation (1), when A is nonsingular, in other words the matrix pth roots that are expressible as polynomials in A. Such solutions are polynomial functions of a matrix, known as primary matrix functions (see [11], [13] and [16]). ...
... In general, a matrix pth root may not exist or there may be an infinite number of solutions for (1). In this study, we are particularly interested in the polynomial solutions of Equation (1), when A is nonsingular, in other words the matrix pth roots that are expressible as polynomials in A. Such solutions are polynomial functions of a matrix, known as primary matrix functions (see [11], [13] and [16]). The function considered here is nothing else but only the complex pth root function f (z) ≡ z 1/p , which is a multi-valued function. ...
Preprint
This study is devoted to the polynomial representation of the matrix pth root functions. The Fibonacci-H\"orner decomposition of the matrix powers and some techniques arisen from properties of generalized Fibonacci sequences, notably the Binet formula, serves as a triggering factor to provide explicit formulas for the matrix pth roots. Special cases and illustrative numerical examples are given.
... Some extensions of the spectral properties of the TT matrices are also pointed-out. For instance, if a block-TT matrix of this kind is also diagonalizable, and certain matrix square-root [33][34][35] exist, such block-TT matrix and a related block-TST one are isospectral matrices. The eigenvalues of proper block-TT matrices are analyzed in Section 3 , and the previous isospectral property is extended to any block-TT matrix with commuting matrix-entries. ...
... Since the matrix-entries C and B are non-singular, the matrix square-root [33][34][35] of the non-singular matrix C −1 B is well defined. The matrix √ C −1 B is also non-singular and it should computed in the appropriate branch. ...
... We are interested in the eigenvalues of T . The matrix product BC is singular, but it has primary matrix square-roots [34,35] . Taking the appropriate branch, ...
Article
After a short overview, improvements (based on the Kronecker product) are proposed for the eigenvalues of (N × N) block-Toeplitz tridiagonal (block-TT) matrices with (K × K) matrix-entries, common in applications. Some extensions of the spectral properties of the Toeplitz-tridiagonal matrices are pointed-out. The eigenvalues of diagonalizable symmetric and skew-symmetric block-TT matrices are studied. Besides, if certain matrix square-root is well-defined, it is proved that every block-TT matrix with commuting matrix-entries is isospectral to a related symmetric block-TT one. Further insight about the eigenvalues of hierarchical Hermitian block-TT matrices, of use in the solution of PDEs, is also achieved.
... For example, this matrix is involved in control and systems theory, matrix differential equations, nonlinear matrix equations, finance and health care. Many methods and techniques have been expanded to provide exact and approximate representations of the matrix pth root (see [1], [4], [9], [17], and references therein). In this study, we consider the Fibonacci-Hörner decomposition of the matrix powers (see [2], [5], [6] and [7]) and some techniques based on some properties of generalized Fibonacci sequences (see [10] and [18]), to provide some explicit formulas of the matrix pth roots. ...
... A matrix pth root can be defined using several definitions of a matrix function of the current literature (see [11], [12], [13], [16] and [19]). In general, a matrix pth root may not exist or there may be an infinite number of solutions for (1). In this study, we are particularly interested in the polynomial solutions of Equation (1), when A is nonsingular, in other words the matrix pth roots that are expressible as polynomials in A. Such solutions are polynomial functions of a matrix, known as primary matrix functions (see [11], [13] and [16]). ...
... In general, a matrix pth root may not exist or there may be an infinite number of solutions for (1). In this study, we are particularly interested in the polynomial solutions of Equation (1), when A is nonsingular, in other words the matrix pth roots that are expressible as polynomials in A. Such solutions are polynomial functions of a matrix, known as primary matrix functions (see [11], [13] and [16]). The function considered here is nothing else but only the complex pth root function f (z) ≡ z 1/p , which is a multi-valued function. ...
Article
This study is devoted to the polynomial representation of the matrix pth root functions. The Fibonacci-H\"orner decomposition of the matrix powers and some techniques arisen from properties of generalized Fibonacci sequences, notably the Binet formula, serves as a triggering factor to provide explicit formulas for the matrix pth roots. Special cases and illustrative numerical examples are given.
... Dans le cas général, une racine p-ième de la matrice A peut ne pas exister comme il peut y avoir une infinité de solutions de (1). Notons qu'une matrice inversible A admet toujours une racine p-ième (ou plus). ...
... Dans les articles [1] et [10], nousétablissons deux formules explicites pour calculer les racines p-ièmes d'une matrice. Cette approche fait intervenir les suites de Fibonacci généralisées, par le moyen de la formule de Binet, employée dans la représentation polynomiale des puissances n-ièmes d'une matrice. ...
... Dans l'article [1], nousétablissons une formulation polynomiale de la racine p-ième principale d'une matrice, valable pour des matrices dont le spectre est inclu dans C\R − . ...
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... such that the ρ(n + r, r) are given by (2). It was shown that φ(t) satisfies the following ordinary differential equation y (r) (t) = a 0 y (r−1) (t) + a 1 y (r−2) (t) + · · · + a r−1 y(t). ...
... We We have hn = ∑︀ (︃ |j| j 1 , j 2 , · · · , jm )︃ (−1) |j| b j1 1 b j2 2 · · · b jm m , where the summation runs over the multi indices j = (j 1 , j 2 , · · · , jm) with nonnegative coordinates such that α(j) = k, where α(j) = j 1 + 2j 2 + · · · + mjm, and |j| = j 1 + j 2 + · · · + jm (for more details, see proof of Proposition 4.4 in [23]). When we take ω(z) = P A (z), we derive that the sequence for hn is nothing else but the Fibonacci sequence ρ(n + r, r) defined by Expression (2). Yet, another formulation of hn in terms of the roots of P A (z) is provided in Corollary 4.4 of [23]. ...
... For purpose of illustration, we examine the following example. Recently, some explicit formulas of the principal matrix pth root have been obtained in [2,8]. Note that, the Binet formula of sequence (1) and Expression (2) play a central role in the results of [2]. ...
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We present a constructive procedure for establishing explicit formulas of the constituents matrices. Our approach is based on the tools and techniques from the theory of generalized Fibonacci sequences. Some connections with other results are supplied. Furthermore,we manage to provide tractable expressions for the matrix functions, and for illustration purposes we establish compact formulas for both the matrix logarithm and the matrix pth root. Some examples are also provided.
... Of course, if we set F = exp in (2.1) we recover (1.2). Several other analytic matrix functions such as those considered in [1,21,22] and (1.9) are all special cases of (2.1) with appropriate domains of convergence. In this way, our access to OMC is based on the definition that follows. ...
... We recall that [28, Theorems 1 and 2] gives a representation of the exponential function in (1.2) as a finite matrix sum. Other relevant papers in this direction employing distinct methods and including other matrix functions comprise [8] using the Jordan canonical form and properties of the minimal polynomial of a matrix, [23,36] using the Horner polynomials, [5,6,7,21,22] concerning a combinatorial method based on generalized Fibonacci sequences, and [15] using path-sums. The representation of the analytic matrix function (2.1) as a finite sum is expected from the Cayley-Hamilton theorem which relates A N ∈ C N ×N to lower powers of A. Note that the proof of [28, Theorems 1 and 2] uses the Cayley-Hamilton theorem, but the proof holds if the characteristic polynomial ...
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We show that Putzer's method to calculate the matrix exponential in [28] can be generalized to compute an arbitrary matrix function defined by a convergent power series. The main technical tool for adapting Putzer's formulation to the general setting is the omega matrix calculus; that is, an extension of MacMahon's partition analysis to the realm of matrix calculus and the method in [6]. Several results in the literature are shown to be special cases of our general formalism, including the computation of the fractional matrix exponentials introduced by Rodrigo [30].Our formulation is a much more general, direct, and conceptually simple method forcomputing analytic matrix functions. In our approach the recursive system of equations the base for Putzer's method is explicitly solved, and all we need todetermine is the analytic matrix functions. For more information see https://ejde.math.txstate.edu/Volumes/2021/97/abstr.html
... In general, determining the square root of a matrix A of order d (d ≥ 2), defined as the solution of the matrix equation X 2 = A, is not an easy task. Several studies in the literature are devoted to the square root of matrices (see, for instance, [1,14,16] and references therein). When the matrix A has no eigenvalues on R − (the closed negative real axis), there exists a unique matrix X such that X 2 = A and the eigenvalues of X lies on the segment {z ∈ C : −π/2 < arg(z) < π/2}, where arg(z) is the argument of z (see, for instance, [16] and references therein). ...
... There are several characterizations of the existence and uniqueness of the principal square root of a given matrix (see, for example, [1,14,16] and references therein). An Hermitian matrix A of C d×d , namely, ...
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This paper concerns another approach for solving the matrix differential equations of the second order X(t)=AX(t)+BX(t)X''(t)=AX'(t)+BX(t), where A, B are square matrices of order d×d{d\times d}. Such approach is based on some properties of matrix square root and the linear difference equation. We establish various new results and explicit formulas for the solutions of this type of matrix differential equations. Moreover, because of the non-commutativity condition between A and B, the matrix fundamental system will play an important role. Finally, our results and their robustness, are validated by providing some examples and applications.
... Theorem 4. For any initial point r (0) ∈ 0, 16 9 , the sequence r (k+1) = g(r (k) ) is second order convergent to r = 1, in which the function g(r) is defined by (5). ...
... see e.g. [5]. ...
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Local identifications of piece-wise deterministic models of genetics networks
  • E Cinquemani
  • A Milias-Argeitis
  • J Lygeros
E. Cinquemani, A. Milias-Argeitis, J. Lygeros, Local identifications of piece-wise deterministic models of genetics networks, in: Hybrid Systems: Computation and Control, in: R. Majumdar, P. Tabuada (Eds.), Lecture Notes in Comput. Sci., vol. 5469, Springer, NewYork, 2009, pp. 105-119.
Verde-Star, Functions of matrices
L. Verde-Star, Functions of matrices, Linear Algebra Appl. 406 (2005) 285-300.