特征多项式的Souriau方法 [英] Souriau method for Characteristic Polynomial
问题描述
有人知道用于发现n×n矩阵的特征多项式的Souriau方法吗?我发现第一个系数很明显,但是如何找出其他系数呢?在我需要对矩阵求逆之后,我知道如何.
Does anyone know the Souriau method for finding the characteristic polynomial of any n × n matrix? I found out the first coefficient, is obvious, but how can I find out the other coefficients? After I need to inverse the matrix but I know how.
#include <iostream>
#include <fstream>
using namespace std;
double trace(double a[5][5],int n){
int i;
double trace=0;
for(i=0;i<n;i++)
trace+=a[i][i];
return trace;
}
double prod(double a[5][5],double b[5][5],int n) {
double c[5][5];
int i,j,k;
cout << "\nProd:\n";
for(i=0;i<n;++i){
for(j=0;j<n;++j){
c[i][j]=0;
for(k=0;k<n;++k)
c[i][j]=c[i][j]+(a[i][k]*b[k][j]);
cout << c[i][j] << " ";
}
cout << "\n";
}
return c[i][j];
}
double theta(double a[5][5], int n){
int i;
double theta[5];
theta[1]=-trace(a,n);
for(i=0;i<n;i++)
cout << "Theta[" << i+1 << "]=" << theta[i+1] << "\n";
return theta[i+1];
}
int main(){
ifstream f("a.txt");
ifstream g("b.txt");
double a[5][5],b[5][5];
int i,j,n;
f >> n;
g >> n;
for(i=0;i<n;++i)
for(j=0;j<n;++j)
f >> a[i][j];
cout << "Matrix A:"<<endl;
for(i=0;i<n;++i){
for(j=0;j<n;++j)
cout << a[i][j] << " ";
cout << endl;
}
cout << endl;
for(i=0;i<n;++i)
for(j=0;j<n;++j)
g >> b[i][j];
cout << "Matrix B:" << endl;
for(i=0;i<n;++i){
for(j=0;j<n;++j)
cout << b[i][j] << " ";
cout << endl;
}
cout << endl;
cout << "Trace = ";
cout << trace(a,n);
cout << endl;
prod(a,b,n);
cout << endl;
theta(a,n);
}
推荐答案
来自 https://math.stackexchange.com/a/405975/115115 J. M.
C=A;
for k=1,…,n
if k>1
C=A*(C+c[n−k+1]*I);
c[n−k]=−tr(C)/k;
end for
如果您能读德语,则在 https://de.wikipedia.org/wiki/Algorithmus_von_Faddejew-Leverrier (添加2017年:或同样好的英文版本
If you can read the german, there is a wiki page with extended pseudo-code algortihm at https://de.wikipedia.org/wiki/Algorithmus_von_Faddejew-Leverrier (add 2017: or the equally good english version https://en.wikipedia.org/wiki/Faddeev%E2%80%93LeVerrier_algorithm)
如果您想直接计算逆矩阵,则可以像在Wiki页面中那样,通过C = A B使用与上面的矩阵C相关的矩阵B.如在Wiki页面中可以看到的那样,这给出了稍微复杂一些的算法.但是,最后一个矩阵B满足A B = -c [0] * I,因此,如果有一个逆矩阵,则可以直接计算.
If you want to directly compute the inverse matrix, then you have, as in the wiki page, use the matrix B that is related to the matrix C above via C=AB. This gives, as can be seen in the wiki page, a slightly more complicated algorithm. However, then the last matrix B satisfies AB=-c[0]*I, so that the inverse matrix, if there is one, can be directly computed.
这篇关于特征多项式的Souriau方法的文章就介绍到这了,希望我们推荐的答案对大家有所帮助,也希望大家多多支持IT屋!