习题2.4
生成一个5阶的Hilbert 矩阵，
\[H = (h_{ij})_{n×n} , h_{ij} = \frac{1}{i+j-1} , i,j = 1,2,...,n.\]- 计算Hilbert 矩阵H 的行列式.
- 求H的逆矩阵.
- 求H的特征值和特征向量.
解：
> n=5;x=array(0,dim=c(n,n))
> for(i in 1:n){
+ for(j in 1:n){
+ x[i,j]<-1/(i+j-1)
+ }
+ }
> n
[1] 5
> x
[,1] [,2] [,3] [,4] [,5]
[1,] 1.0000000 0.5000000 0.3333333 0.2500000 0.2000000
[2,] 0.5000000 0.3333333 0.2500000 0.2000000 0.1666667
[3,] 0.3333333 0.2500000 0.2000000 0.1666667 0.1428571
[4,] 0.2500000 0.2000000 0.1666667 0.1428571 0.1250000
[5,] 0.2000000 0.1666667 0.1428571 0.1250000 0.1111111
>
(1):
> det(x)
[1] 3.749295e-12
>
(2):
> solve(x)
[,1] [,2] [,3] [,4] [,5]
[1,] 25 -300 1050 -1400 630
[2,] -300 4800 -18900 26880 -12600
[3,] 1050 -18900 79380 -117600 56700
[4,] -1400 26880 -117600 179200 -88200
[5,] 630 -12600 56700 -88200 44100
>
(3):
> s=crossprod(x,x)
> e=eigen(s)
> e
$values
[1] 2.455648e+00 4.348652e-02 1.301309e-04 9.357361e-08 1.081050e-11
$vectors
[,1] [,2] [,3] [,4] [,5]
[1,] -0.7678547 0.6018715 -0.2142136 0.04716181 0.006173863
[2,] -0.4457911 -0.2759134 0.7241021 -0.43266733 -0.116692747
[3,] -0.3215783 -0.4248766 0.1204533 0.66735044 0.506163658
[4,] -0.2534389 -0.4439030 -0.3095740 0.23302451 -0.767191193
[5,] -0.2098226 -0.4290134 -0.5651934 -0.55759995 0.376245546
>