Linear Algebra and Matrix Analysis for Statistics
Linear Algebra and Matrix Analysis for Statistics
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DOI：
10.1201/b17040
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发表时间：
2014-06
期刊：
影响因子：
--
通讯作者：
Sw. Banerjee;Anindya Roy
中科院分区：
文献类型：
--
作者：
Sw. Banerjee;Anindya Roy
矩阵，向量，基本定义和符号矩阵加法和标量矩阵乘法矩阵乘法分块矩阵方阵的迹一些特殊矩阵线性方程组介绍高斯消元高斯-约当消去法基本矩阵齐次线性系统矩阵的逆关于线性方程组的更多信息LU分解Crout算法LU分解与行交换分块矩阵的LDU分解与Cholesky分解分块矩阵的逆分块矩阵的LDU分解莫里森公式欧几里得空间介绍向量加法和标量乘法线性空间和子空间的交集与子空间的和线性组合和跨度四个基本子空间线性独立基和维数矩阵的秩矩阵的秩和零矩阵的基四个基本子空间的秩和逆秩分解分块矩阵的秩范式使用秩范式的基本子空间的基互补子空间子空间的和子空间的和的维数直接和与补投影子空间、正交子空间和投影内积、范数和正交行秩=列秩：一个利用正交投影的证明施密特正交化正交互补子空间线性代数基本定理更多关于正交性正交矩阵QR分解正交投影和投影仪正交投影仪：替代导子正交投影的和正交三角化重访线性方程组介绍矩阵空间和线性系统的通解秩和线性系统矩阵的广义逆广义逆和线性系统Moore-Penrose逆行列式定义行列式的一些基本性质行列式乘积的行列式计算行列式矩阵转置的行列式-再访分块矩阵的行列式余因子和展开定理矩阵的子式和秩Cauchy-Binet公式拉普拉斯展开式特征值与特征向量特征多项式及其根真实的对称矩阵的谱分解Hermitian矩阵与正规矩阵的谱分解特征值的进一步结果奇异值分解奇异值分解（SVD）奇异值分解（SVD）和四个基本子空间SVD和线性系统SVD，数据压缩和主成分计算奇异值分解Jordan标准形Jordan标准形的含义二次型介绍二次型矩阵正定和非负定矩阵同余和西尔维斯特惯性定律非负定矩阵和未成年人二次型的极值同时对角化克罗内克积及相关运算双线性插值和克罗内克积的基本性质克罗内克积的逆、秩和非奇异性克罗内克积的矩阵因子分解特征值和行列式vec和换位子算子涉及克罗内克积的线性系统西尔维斯特方程和克罗内克和阿达玛积线性迭代系统，范数和收敛线性迭代系统和矩阵幂的收敛向量范数谱半径和矩阵收敛矩阵范数和Gerschgorin圆SVD -重访网页排名和马尔可夫链解线性方程的迭代算法抽象线性代数一般向量空间一般内积线性变换，伴随和秩四个基本子空间--再访线性变换的逆线性变换和矩阵基的变化，等价和相似矩阵希尔伯特空间参考文献练习出现在每章的末尾。
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.