Linear algebra and matrix computations for statistics. (2014)

• Record Type: Book
• Further Information: Note: By Sudipto Banerjee, Anindya Roy.
• Authors: Banerjee, Sudipto; Roy, Anindya, 1970-
• Contents: 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.
• Edition: 1st
• Publisher Details: Boca Raton : Chapman & Hall/CRC
• Publication Date: 2014
• Extent: 1 online resource, illustrations
• Subjects: 519.5; Linear models (Statistics); Algebras, Linear; Matrices; Vector analysis
• Languages: English
• ISBNs: 9781482248265
• Related ISBNs: 9781482248241; 9781439800119
• Notes: Note: Description based on CIP data; resource not viewed.
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