与给定矢量形成正交基的矩阵 [英] matrix that forms an orthogonal basis with a given vector

问题描述

一个线性代数问题;

给定一个k-变量赋范向量u（即u：|| u || _2 = 1）
怎么做你构造单位向量的任意任意k *（k-1）
矩阵，使得（u，\\ gamma_u）形成
正交基？？

我的意思是：从计算机立场的角度来看：
你用什么算法来构造这样的矩阵？

预先感谢， 解决方案

应用u_0的Gram Schmidt正交化和k-1随机生成的向量。如果在某一时刻GS算法产生一个零向量，那么你有一个线性依赖关系，在这种情况下，再次随机选择向量。然而，这种方法是不稳定的，数值很小矢量表示中的错误被放大。然而，这个算法存在一个稳定的修改：

让 a_1 = u，a_2，... a_k 是随机选择的向量

  for i = 1 to k do 
 vi = ai 
 end for 
 
 for i = 1 to k do 
 rii = | vi | 
 qi = vi / rii 
 for j = i + 1 to k do 
 rij =< qi，vj> 
 vj = vj -rij * qi 
 end for 
 end for 
  

结果向量 v1，... vk 将成为矩阵的列，其中 v1 = u 。如果某点 vj 变为零，则选择一个新的矢量 aj 并重新开始。请注意，如果向量a2，..，ak是随机选择的，则发生这种情况的概率可以忽略不计。

A linear algebra question;

Given a k-variate normed vector u (i.e. u : ||u||_2=1) how do you construct \Gamma_u, any arbitrary k*(k-1) matrix of unit vectors such that (u,\Gamma_u) forms an orthogonal basis ?

I mean: from a computationnal stand point of view: what algorithm do you use to construct such matrices ?

Thanks in advance,

解决方案

The naive approach would be to apply Gram Schmidt orthogonalisation of u_0, and k-1 randomly generated vectors. If at some point the GS algorithm generates a zero vector, then you have a linear dependency in which case choose the vector randomly again.

However this method is unstable, small numerical errors in the representation of the vectors gets magnified. However there exists a stable modification of this algorithm:

Let a_1 = u, a_2,...a_k be randomly chosen vectors

for i = 1 to k do 
        vi = ai
end for 

for i = 1 to k do
    rii = |vi| 
    qi = vi/rii
    for j = i + 1 to k do
       rij =<qi,vj>
       vj =vj −rij*qi 
    end for
end for

The resulting vectors v1,...vk will be the columns of your matrix, with v1 = u. If at some point vj becomes zero choose a new vector aj and start again. Note that the probability of this happening is negligible if the vectors a2,..,ak are chosen randomly.

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