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2014 | Buch

Fundamentals of Operators and Matrices Kapitel lesen Erstes Kapitel lesen

Introduction to Matrix Analysis and Applications

verfasst von: Fumio Hiai, Dénes Petz

Verlag: Springer International Publishing

Buchreihe : Universitext

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Matrices can be studied in different ways. They are a linear algebraic structure and have a topological/analytical aspect (for example, the normed space of matrices) and they also carry an order structure that is induced by positive semidefinite matrices. The interplay of these closely related structures is an essential feature of matrix analysis.

This book explains these aspects of matrix analysis from a functional analysis point of view. After an introduction to matrices and functional analysis, it covers more advanced topics such as matrix monotone functions, matrix means, majorization and entropies. Several applications to quantum information are also included.

Introduction to Matrix Analysis and Applications is appropriate for an advanced graduate course on matrix analysis, particularly aimed at studying quantum information. It can also be used as a reference for researchers in quantum information, statistics, engineering and economics.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Fundamentals of Operators and Matrices
Abstract
A linear mapping is essentially a matrix if the vector space is finite-dimensional. In this book the vector space is typically a finite-dimensional complex Hilbert space.
Fumio Hiai, Dénes Petz
Chapter 2. Mappings and Algebras
Abstract
Most of the statements and definitions in this chapter are formulated in the Hilbert space setting.
Fumio Hiai, Dénes Petz
Chapter 3. Functional Calculus and Derivation
Abstract
Let \(A\in \mathbb {M}_n({\mathbb C})\) and \(p(x):=\sum _i c_i x^i\) be a polynomial. It is quite obvious that by \(p(A)\) we mean the matrix \(\sum _i c_i A^i\). So the functional calculus is trivial for polynomials.
Fumio Hiai, Dénes Petz
Chapter 4. Matrix Monotone Functions and Convexity
Abstract
Let \((a, b) \subset {\mathbb R}\) be an interval. A function \(f: (a, b) \rightarrow {\mathbb R}\) is said to be monotone for \(n \times n\) matrices if \(f(A) \le f(B)\) whenever \(A\) and \(B\) are self-adjoint \(n \times n\) matrices, \(A \le B\) and their eigenvalues are in \((a, b)\). If a function is monotone for every matrix size, then it is called matrix monotone or operator monotone.
Fumio Hiai, Dénes Petz
Chapter 5. Matrix Means and Inequalities
Abstract
The study of numerical means has been a popular subject for centuries, and the inequalities
Fumio Hiai, Dénes Petz
Chapter 6. Majorization and Singular Values
Abstract
A citation from von Neumann: “The object of this note is the study of certain properties of complex matrices of \(n\)th order: \(A=(a_{ij})_{i,j=1}^n\), \(n\) being a finite positive integer: \(n=1,2,\dots \). Together with them we shall use complex vectors of \(n\)th order (in \(n\) dimensions): \(x=(x_i)_{i=1}^n\).” This classical subject in matrix theory is exposed in Sects. 6.2 and 6.3 after discussions on vectors in Sect. 6.1. This chapter also contains several matrix norm inequalities as well as majorization results for matrices, which were mostly developed more recently.
Fumio Hiai, Dénes Petz
Chapter 7. Some Applications
Abstract
Matrices are important in many areas of both pure and applied mathematics. In particular, they play essential roles in quantum probability and quantum information.
Fumio Hiai, Dénes Petz
Backmatter
Metadaten
Titel
Introduction to Matrix Analysis and Applications
verfasst von
Fumio Hiai
Dénes Petz
Copyright-Jahr
2014
Electronic ISBN
978-3-319-04150-6
Print ISBN
978-3-319-04149-0
DOI
https://doi.org/10.1007/978-3-319-04150-6

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