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On explicit formulas for the principal matrix logarithm

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Abstract

We describe a method for evaluating both the Fibonacci-Hörner and the polynomial decomposition of the principal matrix logarithm, with a view to solve the lifting problem of its explicit computation. The Binet formula for linear recursive sequences serves as a triggering factor for giving the exact formula. We supply some illustrative examples.

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... 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. ...
... 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. Indeed, for every non-zero complex number z = |z| exp[i arg(z)] (−π < arg(z) ≤ π), it is well known that z admits p pth roots given through the use of the functions f j (z) = |z| 1/p exp(i[arg(z) + 2πj]/p) = z 1/p exp(2iπj/p), j ∈ R(p), (2) where R(p) = {0, 1, . . . , p − 1}. ...
... The matrix power series expansion of the adequate function has been used to study the principal matrix pth root (see [1]) and the principal matrix logarithm (see [2]). One of our main goals is to determine an explicit formula for the principal matrix pth root function g(tA), based mainly on the formula (3) and the Fibonacci-Hörner decomposition. ...
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.
... Our motivation comes from the fact that few papers have considered exhibiting concise formulas for the matrix logarithm. It is important to point out that other methods for computing the matrix logarithm require advanced theory such as matrix square roots, Schur decompositions, and Pad´e approximants (see [1], [2], [5], [6], and references therein). It should be noted that there are few algorithms for computing logarithms of matrices (see, e.g., [3], [7]). ...
... We also note that in the recent paper [6] the authors were interested in developing an exact computation for the principal logarithm of matrices. More precisely, under some conditions on the norm, they exactly compute the principal matrix logarithm, but in this method it is necessary to solve a system of linear recursion equations, determine the Fibonacci-H¨orner decomposition of the matrix, and study the properties of generalized Fibonacci sequences and the corresponding Binet formula. ...
... To illustrate the previous corollary, we consider the same matrix as in [6], ...
... 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. ...
... 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. Indeed, for every non-zero complex number z = |z| exp[i arg(z)] (−π < arg(z) ≤ π), it is well known that z admits p pth roots given through the use of the functions f j (z) = |z| 1/p exp(i[arg(z) + 2πj]/p) = z 1/p exp(2iπj/p), j ∈ R(p), (2) where R(p) = {0, 1, . . . , p − 1}. ...
... The matrix power series expansion of the adequate function has been used to study the principal matrix pth root (see [1]) and the principal matrix logarithm (see [2]). One of our main goals is to determine an explicit formula for the principal matrix pth root function g(tA), based mainly on the formula (3) and the Fibonacci-Hörner decomposition. ...
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.
... Le dernier chapitre concerne la formulation de la racine p-ième principale d'une matrice, a l'aide des approches de Abderramán Marrero et al. [2] et Ben Taher-Rachidi [13], réalisés dans le cas du logarithme principal d'une matrice et fonctions de matrices. ...
... Similarly to the matrix logarithm function (see for example [56], [62], [2], [51]) the definition of the matrix pth root and the principal matrix pth root is not easy to identify, we give here the basic tool for our study. Let A be a real or complex matrix of order r and p ≥ 2 be a positive integer. ...
... k A k have been explored for e tA in [17], [16], [30] and the principal matrix logarithm Log(I −tA) in [2]. We recall that, such expression is called the polynomial decomposition in [2], [17], [16], [30]. ...
Thesis
Full-text available
... Similarly to the matrix logarithm function (see for example [11], [12], [1], [9]) the definition of the matrix pth root and the principal matrix pth root is not easy to identify, we give here the basic tool for our study. Let A be a real or complex matrix of order r and p ≥ 2 be a positive integer. ...
... 2 [4], [8] and the principal matrix logarithm Log(I − tA) in [1]. We recall that, such expression is called the polynomial decomposition in [1], [3], [4], [8]. ...
... 2 [4], [8] and the principal matrix logarithm Log(I − tA) in [1]. We recall that, such expression is called the polynomial decomposition in [1], [3], [4], [8]. Moreover, assume that A, B ∈ M r (C) are two similar matrices, that means there exists a non-singular matrix Z such that B = Z −1 AZ. ...
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In this paper we project to develop two methods for computing the principal matrix pth root. Our approach makes use of the notion of primary matrix functions and minimal polynomial. Therefore, compact formulas for the principal matrix pth root are established and significant cases are explored.
... 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
... j n j λ n i , for n ≥ 1, the dynamic solution of Equation(1), submitted to the prescribed initial condition ρ(0, r) = ρ(1, r) = ... = ρ(r − 1, r) = 0 and ρ(r, r) = 1. Then, we have ...
... , the constituent matrices of A, are given by (11)- (12) and also (4). Note that, some explicit formulas for the principal matrix logarithm have been established in [1], using the Binet formula of sequences (1). Another interesting application is provided by applying Theorem 3.1 for f (z) = p √ z = z 1/p (p ∈ N). ...
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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.
... 2. Power decomposition of A 1 p and its computation Let B be in M r ðCÞ (r P 2) such that P B ðzÞ ¼ z r À a 0 z rÀ1 À Á Á Á À a rÀ1 , with a rÀ1 -0. Following Cayley-Hamilton's theorem, the positive integer powers B n satisfy the rth linear recurrence relation B nþ1 ¼ a 0 B n þ a 1 B nÀ1 þ Á Á Á þ a rÀ1 B nÀrþ1 , for n P r À 1 (see [1,7,8,18]). Thus, the powers of B n (n P r) are given by ...
... As a potential variant for accomplishing explicit formulas of the principal pth root of an n  n matrix A is the introduction of the method used in [1] for calculating the principal logarithm of a matrix. This is based on the Binet formula for the solutions of the generalized Fibonacci sequences; see for instance [11]. ...
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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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On the computation of matrix logarithm
  • N Sherif
  • E Morsy
N. Sherif and E. Morsy, On the computation of matrix logarithm, J. Appl. Math. & Informatics 27 (2009) 105-121.
  • R P Stanley
R. P. Stanley, Enumerative combinatorics, Vol. I, Cambridge University Press, U.K., 1997.