Linear Algebra and Matrix Analysis for Statistics

ISBN-13: 9781420095388 / Angielski / Twarda / 2014 / 582 str.

Sudipto Banerjee;Anindya Roy

Opis

Linear Algebra and Matrix Analysis for Statistics offers a gradual exposition to linear algebra without sacrificing the rigor of the subject. It presents both the vector space approach and the canonical forms in matrix theory. The book is as self-contained as possible, assuming no prior knowledge of linear algebra. The authors first address the rudimentary mechanics of linear systems using Gaussian elimination and the resulting decompositions. They introduce Euclidean vector spaces using less abstract concepts and make connections to systems of linear equations wherever possible. After illustrating the importance of the rank of a matrix, they discuss complementary subspaces, oblique projectors, orthogonality, orthogonal projections and projectors, and orthogonal reduction. The text then shows how the theoretical concepts developed are handy in analyzing solutions for linear systems. The authors also explain how determinants are useful for characterizing and deriving properties concerning matrices and linear systems. They then cover eigenvalues, eigenvectors, singular value decomposition, Jordan decomposition (including a proof), quadratic forms, and Kronecker and Hadamard products. The book concludes with accessible treatments of advanced topics, such as linear iterative systems, convergence of matrices, more general vector spaces, linear transformations, and Hilbert spaces.

Linear algebra and the study of matrix algorithms have become fundamental to the development of statistical models. Using a vector space approach, this book provides an understanding of the major concepts that underlie linear algebra and matrix analysis. Each chapter introduces a key topic such as infinite-dimensional spaces and provides illustrative examples. The author examines recent developments in diverse fields such as spatial statistics, machine learning, data mining and social network analysis. Complete in its coverage and accessible to students without prior knowledge of linear algebra, the text also includes results that are useful for traditional statistical applications.

 

Spis treści

Matrices, Vectors, and Their Operations. Systems of Linear Equations. More on Linear Equations. Euclidean Spaces. The Rank of a Matrix. Complementary Subspaces. Orthogonality, Orthogonal Subspaces, and Projections. More on Orthogonality. Revisiting Linear Equations. Determinants. Eigenvalues and Eigenvectors. Quadratic Forms. The Kronecker Product and Related Operations. Linear Iterative Systems, Norms, and Convergence. Abstract Linear Algebra. References.

O autorze

Sudipto Banerjee, Anindya Roy