title: [线性代数]6-5:正定矩阵(Positive Definite Matrices) categories: Mathematic Linear Algebra keywords: Positive Definite Matrices Symmetric Matrices Eigenvalues Eigenvectors toc: true date: 2017-11-24 11:24:21 Abstract: 关于正定矩阵的相关知识总结,正定矩阵在数学中的一个应用 Keyword…

设M是n阶方阵,如果对任何非零向量z,都有zTMz> 0,其中zT 表示z的转置,就称M正定矩阵. 正定矩阵在合同变换下可化为标准型, 即对角矩阵. 所有特征值大于零的对称矩阵也是正定矩阵.   判定定理1:对称阵A为正定的充分必要条件是:A的特征值全为正. 判定定理2:对称阵A为正定的充分必要条件是:A的各阶顺序主子式都为正. 判定定理3:任意阵A为正定的充分必要条件是:A合同于单位阵.   正定矩阵的性质: 1.正定矩阵的任一主子矩阵也是正定矩阵. 2.若A为n阶对称正定矩阵,则存在唯一的主…

1. 基本定义 在线性规划中,一个对称的 n×n 的实值矩阵 M,如果满足对于任意的非零列向量 z,都有 zTMz>0. 更一般地,对于 n×n 的 Hermitian 矩阵(原矩阵=共轭转置,aij=a¯ji,或者 A=AT¯¯¯¯¯),对于任何的非零列向量 z,z⋆Mz>0: 2. 定理和推论 对称阵 A 为正定的充分必要条件是: A 的特征值全为正: A 的各阶主子式都为正: 对称阵 A 为负定的充分必要条件是:奇数阶主子式为负,偶数阶主子式为正: 3. 正定的几何意义 设 f(x,y)…

https://en.wikipedia.org/wiki/Definite_quadratic_form https://www.math.utah.edu/~zwick/Classes/Fall2012_2270/Lectures/Lecture33_with_Examples.pdf…

搞统计的线性代数和概率论必须精通,最好要能锻炼出直觉,再学机器学习才会事半功倍. 线性代数只推荐Prof. Gilbert Strang的MIT课程,有视频,有教材,有习题,有考试,一套学下来基本就入门了. 不多,一共10次课. 链接:https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/calendar/ SES # TOPICS KEY DATES 1 The geometry of linear e…

I. 范数(Norm) 定义: 向量空间\(V\)上的范数(norm)是如下函数: \[ \begin{align} \|·\|:V→R, \notag \\ x→\|x\| \notag \end{align} \] 该函数会赋予每个向量\(x\)自身的长度\(\|x\|∈R\),并且对于\(\lambda∈R,\,\,x,y∈V\)满足如下性质: Absolutely homogeneous:\(\|\lambda x\|=|\lambda|\|x\|\) Triangle inequali…

8.02  Physics II (电磁学基础) Introduction to electromagnetism and electrostatics: electric charge, Coulomb's law, electric structure of matter; conductors and dielectrics. Concepts of electrostatic field and potential, electrostatic energy. Electric curr…

IEEE International Conference on Computer Vision, ICCV 2017, Venice, Italy, October 22-29, 2017. IEEE Computer Society 2017, ISBN 978-1-5386-1032-9 Oral Session 1 Globally-Optimal Inlier Set Maximisation for Simultaneous Camera Pose and Feature Corre…

Accepted Papers     Title Primary Subject Area ID 3D computer vision 93 UPnP: An optimal O(n) solution to the absolute pose problem with universal applicability 128 Video Registration to SfM Models 168 Image-based 4-d Modeling Using 3-d Change Detect…

ICLR 2013 International Conference on Learning Representations May 02 - 04, 2013, Scottsdale, Arizona, USA ICLR 2013 Workshop Track Accepted for Oral Presentation Zero-Shot Learning Through Cross-Modal Transfer Richard Socher, Milind Ganjoo, Hamsa Sr…