1st Edition

# Linear Algebra and Matrix Analysis for Statistics

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**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.

**Matrices, Vectors, and Their Operations**Basic definitions and notations

Matrix addition and scalar-matrix multiplication

Matrix multiplication

Partitioned matrices

The "trace" of a square matrix

Some special matrices

**Systems of Linear Equations**Introduction

Gaussian elimination

Gauss-Jordan elimination

Elementary matrices

Homogeneous linear systems

The inverse of a matrix

**More on Linear Equations**The

*LU*decomposition

Crout’s Algorithm

*LU*decomposition with row interchanges

The

*LDU*and Cholesky factorizations

Inverse of partitioned matrices

The

*LDU*decomposition for partitioned matrices

The Sherman-Woodbury-Morrison formula

**Euclidean Spaces**Introduction

Vector addition and scalar multiplication

Linear spaces and subspaces

Intersection and sum of subspaces

Linear combinations and spans

Four fundamental subspaces

Linear independence

Basis and dimension

**The Rank of a Matrix**Rank and nullity of a matrix

Bases for the four fundamental subspaces

Rank and inverse

Rank factorization

The rank-normal form

Rank of a partitioned matrix

Bases for the fundamental subspaces using the rank normal form

**Complementary Subspaces**Sum of subspaces

The dimension of the sum of subspaces

Direct sums and complements

Projectors

**Orthogonality, Orthogonal Subspaces, and Projections**Inner product, norms, and orthogonality

Row rank = column rank: A proof using orthogonality

Orthogonal projections

Gram-Schmidt orthogonalization

Orthocomplementary subspaces

The fundamental theorem of linear algebra

**More on Orthogonality**Orthogonal matrices

The

*QR*decomposition

Orthogonal projection and projector

Orthogonal projector: Alternative derivations

Sum of orthogonal projectors

Orthogonal triangularization

**Revisiting Linear Equations**Introduction

Null spaces and the general solution of linear systems

Rank and linear systems

Generalized inverse of a matrix

Generalized inverses and linear systems

The Moore-Penrose inverse

**Determinants**Definitions

Some basic properties of determinants

Determinant of products

Computing determinants

The determinant of the transpose of a matrix — revisited

Determinants of partitioned matrices

Cofactors and expansion theorems

The minor and the rank of a matrix

The Cauchy-Binet formula

The Laplace expansion

**Eigenvalues and Eigenvectors**Characteristic polynomial and its roots

Spectral decomposition of real symmetric matrices

Spectral decomposition of Hermitian and normal matrices

Further results on eigenvalues

Singular value decomposition

**Singular Value and Jordan Decompositions **Singular value decomposition (SVD)

The SVD and the four fundamental subspaces

SVD and linear systems

SVD, data compression and principal components

Computing the SVD

The Jordan canonical form

Implications of the Jordan canonical form

**Quadratic Forms**Introduction

Quadratic forms

Matrices in quadratic forms

Positive and nonnegative definite matrices

Congruence and Sylvester’s law of inertia

Nonnegative definite matrices and minors

Extrema of quadratic forms

Simultaneous diagonalization

**The Kronecker Product and Related Operations **Bilinear interpolation and the Kronecker product

Basic properties of Kronecker products

Inverses, rank and nonsingularity of Kronecker products

Matrix factorizations for Kronecker products

Eigenvalues and determinant

The vec and commutator operators

Linear systems involving Kronecker products

Sylvester’s equation and the Kronecker sum

The Hadamard product

**Linear Iterative Systems, Norms, and Convergence **Linear iterative systems and convergence of matrix powers

Vector norms

Spectral radius and matrix convergence

Matrix norms and the Gerschgorin circles

SVD – revisited

Web page ranking and Markov chains

Iterative algorithms for solving linear equations

**Abstract Linear Algebra **General vector spaces

General inner products

Linear transformations, adjoint and rank

The four fundamental subspaces - revisited

Inverses of linear transformations

Linear transformations and matrices

Change of bases, equivalence and similar matrices

Hilbert spaces

**References**

Exercises appear at the end of each chapter.

### Biography

Sudipto Banerjee, Anindya Roy

"… a unique and remarkable book … has much to offer that is not found elsewhere. … In

Linear Algebra and Matrix Analysis for Statistics, Sudipto Bannerjee and Anindya Roy have raised the bar for textbooks in this genre. For me, this book will be an invaluable resource for my teaching and research. … an outstanding choice for research-oriented statisticians who want a comprehensive theoretical treatment of the subject that will take them well beyond the prerequisites for the study of linear models."

—Journal of the American Statistical Association, Vol. 110, 2015"The sixteen chapters cover the full range of topics … Topics are presented in a logical order and in a reasonable pace. The book is compactly written and the approach throughout is rigorous, yet well readable. … an excellent introduction to linear algebra."

—Zentralblatt MATH1309"This would be a reasonable candidate for use in a standard linear algebra course, even at institutions with no statistics majors. … The proofs are very detailed and the authors bind the argument together with clear text that flows beautifully. … Some linear algebra courses put a greater emphasis on concrete applications or on using software to get computations done. Other texts treat linear algebra as a branch of abstract algebra and allow spaces over arbitrary fields. This book is a strong contender for the vast majority of linear algebra courses that fall between those two extremes."

—MAA Reviews, October 2014"This beautifully written text is unlike any other in statistical science. It starts at the level of a first undergraduate course in linear algebra, and takes the student all the way up to the graduate level, including Hilbert spaces. It is extremely well crafted and proceeds up through that theory at a very good pace. The book is compactly written and mathematically rigorous, yet the style is lively as well as engaging. This elegant, sophisticated work will serve upper-level and graduate statistics education well. All and all a book I wish I could have written."

—Jim Zidek, University of British Columbia, Vancouver, Canada