一种矩阵运算方法,又叫Cholesky分解.所谓平方根法,就是利用对称正定矩阵的三角分解得到的求解对称正定方程组的一种有效方法.它是把一个对称正定的矩阵表示成一个下三角矩阵L和其转置的乘积的分解.它要求矩阵的所有特征值必须大于零,故分解的下三角矩阵的对角元也是大于零的. https://en.wikipedia.org/wiki/Positive-definite_matrix In linear algebra, a symmetric {\displaystyle n} × {\displa…
降维(Dimensionality Reduction) 动机一:数据压缩(Motivation I : Data Compression) 数据压缩允许我们压缩数据,从而使用较少的计算机内存或磁盘空间,还会加快算法的学习速度. 下面举例说明下降维是什么? 在工业上,往往有成百上千个特征.比如,可能有几个不同的工程团队,一个团队给了你二百个特征,第二个团队给了你另外三百个的特征,第三团队给了你五百个特征,一千多个特征都在一起,那么实际上,如果你想去追踪一下你所知道的那些特征会变得相当困难,而你又…
In linear algebra, a symmetric n × n real matrix M is said to be positive definite if zTMz is positive for every non-zero columnvector z of n real numbers. Here zT denotes thetranspose of z. The real symmetric matrix is positive definite since for an…
I find it may cost me so much time in doing such solutions to exercises and problems....I am sorry that I could not be persistent in doing it...Wish I could just recover it later on. [Bhatia.Matrix Analysis.Solutions to Exercises and Problems]PrI.6.1…
目录:Matrix Differential Calculus with Applications in Statistics and Econometrics,3rd_[Magnus2019] Title -16 Contents -14 Preface -6 Part One - Matrices 1 1 Basic properties of vectors and matrices 3 1.1 Introduction 3 1.2 Sets 3 1.3 Matrices: additio…
Show that the following statements are equivalent: (1). $A$ is positive. (2). $A=B^*B$ for some $B$. (3). $A=T^*T$ for some upper triangular $T$. (4). $A=T^*T$ for some upper triangular $T$ with nonnegative diagonal entries. If $A$ is positive defini…
三对角线性方程组(tridiagonal systems of equations) 三对角线性方程组,对于熟悉数值分析的同学来说,并不陌生,它经常出现在微分方程的数值求解和三次样条函数的插值问题中.三对角线性方程组可描述为以下方程组: \[a_{i}x_{i-1}+b_{i}x_{i}+c_{i}x_{i+1}=d_{i}\] 其中\(1\leq i \leq n, a_{1}=0, c_{n}=0.\) 以上方程组写成矩阵形式为\(Ax=d\),即: \[ {\begin{bmatrix…
当参数 A 是正定矩阵(positive definite)时,logdet 利用相关矩阵分解的性质,将比 log(det(A)) 获得更快的效率: function y = logdet(A) try U = chol(A); y = 2*sum(log(diag(U))) ; catch y = 0; warning('logdet:postdef', 'Matrix is not positive definite'); end end…
There are a number of algorithms that are typically used for system identification, adaptive control, adaptive signal processing, and machine learning. These algorithms all have particular similarities and differences. However, they all need to proce…
http://blogs.mathworks.com/loren/2007/03/01/creating-sparse-finite-element-matrices-in-matlab/ Loren on the Art of MATLAB March 1st, 2007 Creating Sparse Finite-Element Matrices in MATLAB I'm pleased to introduce Tim Davis as this week's guest blogge…
In recent years, Kernel methods have received major attention, particularly due to the increased popularity of the Support Vector Machines. Kernel functions can be used in many applications as they provide a simple bridge from linearity to non-linear…
