Advanced Linear Algebra
Advanced Linear Algebra
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DOI：
10.1201/b18994
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发表时间：
2015-12
期刊：
影响因子：
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通讯作者：
Hugo J. Woerdeman
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文献类型：
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作者：
Hugo J. Woerdeman
场和矩阵代数场Z3场公理场例子不同场上的矩阵代数向量空间向量空间的定义函数子空间的向量空间和更多的向量空间的例子线性无关性、跨度和基坐标系练习线性变换的定义线性变换的范围和核的定义线性映射的矩阵表示练习幂零矩阵的Jordan标准形Cayley-Hamilton定理Jordan标准形关于多项式的Jordan标准形最小多项式交换矩阵矩阵线性微分方程组矩阵的函数的Jordan标准形预解练习内积和赋范向量空间的内积、正交范数集和基Schur三角化正规和厄米特矩阵奇异值分解练习从给定的空间构造新的向量空间笛卡尔乘积，商空间，对偶空间多线性映射和泛函，张量积反对称张量和对称张量练习如何使用线性代数矩阵你不能写下来，但仍然想使用基于矩阵向量积的算法为什么在计算多项式的根时使用矩阵？如何用线性代数求函数？如何处理不完全矩阵用线性代数解决千禧年奖问题RSA加密有多安全？量子计算和正图练习如何开始自己的研究项目练习答案
Fields and Matrix Algebra The Field Z3 The Field Axioms Field Examples Matrix Algebra over Different Fields Exercises Vector Spaces Definition of a Vector Space Vector Spaces of Functions Subspaces and More Examples of Vector Spaces Linear Independence, Span, and Basis Coordinate Systems Exercises Linear Transformations Definition of a Linear Transformation Range and Kernel of Linear Transformations Matrix Representations of Linear Maps Exercises The Jordan Canonical Form The Cayley-Hamilton Theorem Jordan Canonical Form for Nilpotent Matrices An Intermezzo about Polynomials The Jordan Canonical Form The Minimal Polynomial Commuting Matrices Systems of Linear Differential Equations Functions of Matrices The Resolvent Exercises Inner Product and Normed Vector Spaces Inner Products and Norms Orthogonal and Orthonormal Sets and Bases The Adjoint of a Linear Map Unitary Matrices, QR, and Schur Triangularization Normal and Hermitian Matrices Singular Value Decomposition Exercises Constructing New Vector Spaces from Given Ones The Cartesian Product The Quotient Space The Dual Space Multilinear Maps and Functionals The Tensor Product Anti-Symmetric and Symmetric Tensors Exercises How to Use Linear Algebra Matrices You Can't Write Down, but Would Still Like to Use Algorithms Based on Matrix Vector Products Why Use Matrices When Computing Roots of Polynomials? How to Find Functions with Linear Algebra? How to Deal with Incomplete Matrices Solving Millennium Prize Problems with Linear Algebra How Secure Is RSA Encryption? Quantum Computation and Positive Maps Exercises How to Start Your Own Research Project Answers to Exercises