Matrix Methods and Fractional Calculus

Arak M Mathai, Hans J Haubold

• Publisher: WSPC
• Year: 2017
• eBook ISBN: 9789813227545
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

-->

Fractional calculus in terms of mathematics and statistics and its applications to problems in natural sciences is NOT yet part of university teaching curricula. This book is one attempt to provide an approach to include topics of fractional calculus into university curricula. Additionally the material is useful for people who do research work in the areas of special functions, fractional calculus, applications of fractional calculus, and mathematical statistics.

--> Contents:

• Preface
• List of Symbols
• Vector/Matrix Derivatives and Optimization
• Jacobians of Matrix Transformations and Functions of Matrix Argument
• Fractional Calculus and Special Functions
• Fractional Calculus and Fractional Differential Equations
• Kober Fractional Calculus and Matrix-Variate Functions
• Lie Theory and Special Functions
• Selected Topics in Multivariate Analysis

-->
--> Readership: Graduate students and researchers in all aspects of fractional calculus and its applications. -->
Keywords:Vector/Matrix Derivatives;Optimization;Jacobians of Matrix Transformations;Multivariate Analysis;Functions of Matrix Argument;Fractional Calculus;Special Functions;Lie Theory;Fractional Differential Equations;Kober Fractional Calculus;Matrix-Variate FunctionsReview:0

Chapter 1

Vector/Matrix Derivatives and Optimization^∗

1.1.Introduction

The type of functions that we are going to deal with in the present series of lectures is real-valued scalar functions where the argument of the function can be a scalar variable, a vector variable or a matrix variable. To start with, we will consider the argument to be real, real scalar variable or real matrix variable in the sense that the elements of the matrix are real variables or real constants, not complex variables. Later we will consider the situations where the argument is a matrix defined over the complex domain.

The following standard notations will be used: Matrices will be denoted by capital letters, variable matrices by X, Y, Z, . . . and constant matrices by A, B, C, . . . scalar variables by x, y, z, . . . and scalar constants by a, b, . . . . If X = (x_ij) is a p × p matrix then tr(X) = trace of X = sum of the leading diagonal elements = x₁₁ + x₂₂ + ··· + x_pp = λ₁ + λ₂ ··· + λ_p where λ₁, λ₂, . . . , λ_p are the eigenvalues of X. The determinant of X will be denoted by |X| or det(X). For example, if the determinant is a + ib = det(X) where, a, b real and i =

then the conjugate is a − ib and the absolute value is

When f(X) is a real-valued scalar function of the m × n matrix X, then ∫_X f(X)dX will mean the integral over all m × n matrices. Here dX stands for the wedge product of differentials, that is,

where ∧ = wedge, dx ∧ dy is the wedge product or skew symmetric product of differentials dx and dy and the skew symmetric product is defined as

Let us see the consequence of this definition of wedge product. Let f₁(x₁, x₂) and f₂(x₁, x₂) be two real-valued scalar functions of the real scalar variables x₁ and x₂. Let y₁ = f₁(x₁, x₂) and y₂ = f₂(x₁, x₂). Then from basic calculus, the differentials of y₁ and y₂ are given by

Taking wedge product, we have

Straight multiplication of the right side of (iii), keeping the order, gives

The coefficient is the determinant of the matrix

where the (i, j)th element is the partial derivative of f_i with respect to x_j. From the structu...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Preface
• Acknowledgment
• Contents
• List of Symbols
• Chapter 1. Vector/Matrix Derivatives and Optimization
• Chapter 2. Jacobians of Matrix Transformations and Functions of Matrix Argument
• Chapter 3. Fractional Calculus and Special Functions
• Chapter 4. Fractional Calculus and Fractional Differential Equations
• Chapter 5. Kober Fractional Calculus and Matrix-Variate Functions
• Chapter 6. Lie Theory and Special Functions
• Chapter 7. Selected Topics in Multivariate Analysis 247
• Author Index
• Subject Index