矩阵类的要求：
- 设计一个二维矩阵类
- 成员变量为double*类型，用来存储矩阵中的元素
- 写出赋值运算符重载(包括+=，-=，*=)，拷贝构造函数和构造函数
- 实现矩阵的转置操作
- 实现矩阵的逆求解
- 实现矩阵的行列式运算
- 矩阵的加减乘操作
- 利用（）运算符重载可以改变某个矩阵元素的值
- 实现AX=b求解操作
- 求矩阵的秩
- 求矩阵的特征值和特征向量
#include<stdio.h>
#include<cstdio>
#include<iostream>
#include<math.h>
#include<cmath>
#include <stdlib.h>
using namespace std;
const double EPS = 1e-10;
class Matrix{
public:
int row;
int col;
int quantity;
double *matrix;
Matrix(int r,int c):row(r),col(c)//构造函数
{
quantity = r*c;
if (quantity>0)
{
matrix= new double[quantity];
}
else
matrix= NULL;
};
Matrix(const Matrix &rhs)//拷贝构造
{
row = rhs.row;
col = rhs.col;
quantity = rhs.quantity;
matrix = new double[quantity];
for (int i = 0; i<quantity; i++)
matrix[i] = rhs.matrix[i];
}
virtual ~Matrix();
void initmatrix();
Matrix inverse();//矩阵的逆
int rank(const Matrix&);
Matrix &operator=(const Matrix&);
double& operator()(int m,int n);//利用（）运算符重载可以改变某个矩阵元素的值
friend Matrix solveAb(Matrix &,Matrix &);
//friend Matrix operator=(Matrix &,Matrix &);
friend Matrix operator/(const Matrix&, double);
friend istream &operator>>(istream&, Matrix&);
friend ostream &operator<<(ostream&, Matrix&);
friend Matrix operator+(const Matrix&, const Matrix&);
friend Matrix operator-(const Matrix&, const Matrix&);
friend Matrix operator*(const Matrix&, const Matrix&);
friend Matrix operator*(double, const Matrix&); //数乘矩阵
friend Matrix operator*(const Matrix&, double); //矩阵乘数
Matrix operator=(double *);
Matrix& operator+=(const Matrix &m1);
Matrix& operator-=(const Matrix &);
Matrix& operator*=(const Matrix &m1);
double*operator[](int i){ return matrix + i*col; }
Matrix transpose()const;//矩阵转置
double determinant();//矩阵行列式
Matrix Adjugate();
void swapRows(int, int);
Matrix gaussianEliminate();//高斯消元法 const Matrix &m
Matrix diag();
void QR(Matrix&, Matrix&)const;
Matrix eig_val(int _iters = 1000);
Matrix eig_vect(int _iters = 1000);
friend void menu();
};
Matrix::~Matrix()//析构函数
{
for(int i=0;i<row*col;i++)
delete[] matrix;
}
void Matrix::initmatrix()//初始化矩阵
{
int i,j;
matrix=new double[row];
for(i=0;i<row*col;i++)
matrix[i]=0.00;
}
double& Matrix::operator()(int m,int n)
{
return *(matrix+m*col+n);
}
Matrix& Matrix::operator=(const Matrix& rhs)
{
if (this!=&rhs)
{
row=rhs.row;
col=rhs.col;
quantity=rhs.quantity;
if (matrix!=NULL)
delete[] matrix;
matrix=new double[quantity];
for (int i=0;i<quantity;i++)
{
matrix[i]=rhs.matrix[i];
}
}
//cout<<"ok";
return *this;
}
istream& operator>>(istream &is, Matrix &obj)
{
for(int i=0;i<obj.row*obj.col;i++)
{
is>>obj.matrix[i];
}
return is;
}
ostream& operator<<(ostream &out, Matrix &obj)
{
for (int i=0;i<obj.row;i++) //打印矩阵
{
for(int j=0;j<obj.col;j++)
{
out<<(obj[i][j])<<"\t";
}
out<<endl;
}
return out;
}
Matrix operator+(const Matrix& m1,const Matrix& m2)
{
if (m1.col!=m2.col||m1.row!=m2.row)
{
Matrix temp(0, 0);
temp.matrix=NULL;
cout <<"矩阵不合法"<<endl;
return temp;
}
Matrix ret(m1.row, m1.col);
for (int i=0;i<ret.quantity;i++)
{
ret.matrix[i]=m1.matrix[i]+m2.matrix[i];
}
return ret;
}
Matrix operator-(const Matrix &m1, const Matrix &m2)
{
if(m1.row!=m2.row||m1.col!=m1.col)
{
Matrix temp(0,0);
cout<<"矩阵不合法"<<endl;
return temp;
}
Matrix te(m1.row,m1.col);
for(int i=0;i<m1.row*m1.col;i++)
te.matrix[i]=m1.matrix[i]-m2.matrix[i];
return te;
}
Matrix operator*(const Matrix& m1, const Matrix& m2)
{
if (m1.quantity==0||m2.quantity==0||m1.col!=m2.row)
{
Matrix temp(0,0);
temp.matrix=NULL;
cout<<"矩阵不合法"<<endl;
return temp; //数据不合法时候，返回空矩阵
}
Matrix ret(m1.row,m2.col);
for (int i=0;i<m1.row;i++)
{
for (int j=0;j<m2.col;j++)
{
for (int k=0;k<m1.col;k++)//m1.col == m2.row
{
ret.matrix[i*m2.col+j]+=m1.matrix[i*m1.col+k]*m2.matrix[k*m2.col+j];
}
}
}
return ret;
}
Matrix operator*(double val, const Matrix& m) //矩阵乘 单数
{
Matrix ret(m.row, m.col);
for (int i=0;i<ret.quantity;i++)
{
ret.matrix[i]=val*m.matrix[i];
}
return ret;
}
Matrix operator*(const Matrix&m,double val) //矩阵乘 单数
{
Matrix ret(m.row,m.col);
for (int i=0;i<ret.quantity;i++)
{
ret.matrix[i]=val*m.matrix[i];
}
return ret;
}
Matrix &Matrix::operator+=(const Matrix &m1)
{
int i;
for(i=0;i<m1.col*m1.row;i++)
matrix[i]=matrix[i]+m1.matrix[i];
return *this;
}
Matrix &Matrix::operator-=(const Matrix &m1)
{
int i;
for(i=0;i<m1.col*m1.row;i++)
matrix[i]=matrix[i]-m1.matrix[i];
return *this;
}
Matrix &Matrix::operator*=(const Matrix &m)
{
int i,j,k;
if(col==0||row==0||m.col==0||row==0||col!=m.row)
{
Matrix temp(0,0);
cout<<"矩阵不合法"<<endl;
return *this;
}
Matrix te(row,m.col);
for (i=0;i<row;i++)
{ for (j=0;j<m.col;j++)
{ for (k=0;k<col;k++)
{
te.matrix[i*m.col+j]+=matrix[i*col+k]*m.matrix[k*m.col+j];
}
}
}
return *this;
}
Matrix Matrix::transpose()const//实现矩阵的转置操作
{
int k=0;
Matrix tem(col,row);
for (int i=0;i<row;i++)
{
for (int j=0;j<col;j++)
{
tem[j][i]=matrix[i*col+j];
}
}
return tem;
}
double calcDet(int n,double *&aa)
{
if (n==1)
return aa[0];
double *bb=new double[(n-1)*(n-1)];
double sum=0.00;
for (int Ai=0;Ai<n;Ai++)
{
for (int Bi=0;Bi<n-1;Bi++)
{
int offset= Bi<Ai?0:1;
for (int j=0;j<n-1;j++)
{
bb[Bi*(n-1)+j]=aa[(Bi+offset)*n+j+1];
}
}
int flag=(Ai%2==0?1:-1);
sum+= flag*aa[Ai*n]*calcDet(n-1,bb);
}
delete[]bb;
return sum;
}
double Matrix::determinant()
{
if (col==row)
return calcDet(row,matrix);
else
{
cout<<"矩阵不合法"<<endl;
return 0;
}
}
double AlgCofactor(Matrix& mt, int m, int n)//代数余子式
{
int trow=mt.row-1;
Matrix temp(trow,trow);
for(int i=0;i<trow;i++)
for(int j=0;j<trow;j++)
{
int temrow=i<m?0:1;
int temcol=j<n?1:1;
temp[i][j]=mt[i + temrow][j + temcol];
}
int flag;
flag=(m+n)%2==0?1:-1;
return flag*temp.determinant();
}
Matrix Matrix::Adjugate()//伴随矩阵
{
int i,j,k;
if(col!=row)
{
Matrix tem(0,0);
cout<<"矩阵不合法"<<endl;
return tem;
}
Matrix temp(row,col);
for(i=0;i<row;i++)
for(j=0;j<col;j++)
temp.matrix[j*row+i]= AlgCofactor(*this,i,j);
return temp;
}
Matrix operator/(const Matrix& m1, double n) //矩阵除以单数
{
Matrix ret(m1.row,m1.col);
for (int i=0;i<ret.row*ret.col;i++)
{
ret.matrix[i]= m1.matrix[i]/n;
}
return ret;
}
Matrix Matrix::inverse()
{
double det=determinant();
if (det==0)
{
cout << "行列式为0，不能计算逆矩阵。" << endl;
return Matrix(0,0);
}
return Adjugate()/det;
}
Matrix solveAb(Matrix &m1,Matrix &m2)
{
if(m1.row==0||m1.col==0||m2.row==0||m2.col==0)
{
Matrix tem(0,0);
cout<<"矩阵不合法"<<endl;
return tem;
}
Matrix invA(m1.col,m1.row);
invA+=m1.inverse();
Matrix result(invA.row,m2.col);
result+=invA*m2;
return invA;
}
//实现行变换
void Matrix::swapRows(int a, int b)
{
a--;
b--;
double temp = matrix[a];
matrix[a] = matrix[b];
matrix[b] = temp;
}
Matrix Matrix::gaussianEliminate()
{
Matrix Ab(*this);
int rows=Ab.row;
int cols=Ab.col;
int Acols= cols-1;
int i=0; //跟踪行
int j=0; //跟踪列
while (i<rows)
{
bool flag=false;
while (j<Acols&&!flag)
{
if (Ab[i][j]!=0) {
flag=true;
}
else {
int max_row=i;
double max_val=0;
for (int k=i+1;k<rows;++k)
{
double cur_abs=Ab[k][j]>=0?Ab[k][j]:-1*Ab[k][j];
if (cur_abs>max_val)
{
max_row=k;
max_val=cur_abs;
}
}
if (max_row!=i){
Ab.swapRows(max_row, i);
flag=true;
}
else {
j++;
}
}
}
if (flag)
{
for (int t=i+1;t<rows;t++) {
for (int s=j+1;s<cols;s++) {
Ab[t][s]=Ab[t][s]-Ab[i][s]*(Ab[t][j]/Ab[i][j]);
if (abs(Ab[t][s])<1e-10)
Ab[t][s] = 0;
}
Ab[t][j]=0;
}
}
i++;
j++;
}
//cout<<Ab;
return Ab;
}
int Matrix::rank(const Matrix &m)
{
int i,j;
int num,count;
Matrix temp(m.row,m.col);
temp=m;count=0;
//cout<<"阶梯矩阵:\n"<<temp;
for(i=0;i<m.row;i++)
{
num=0;
for(j=0;j<m.col;j++)
{
if(temp[i][j]==0) num++;
}
if(num!=temp.col)
count++;
}
return count;
}
void Matrix::QR(Matrix &Q, Matrix &R) const
{
if (row!= col)
{
printf("矩阵不合法\n");
return;
}
const int N=row;
double *a=new double[N];
double *b=new double[N];
for(int j=0;j<N;++j)
{
for(int i=0; i<N;++i)
a[i]=b[i]=matrix[i*N+j];
for(int i=0;i<j;++i)
{
R.matrix[i*N+j]=0;
for(int m=0;m<N;++m)
{
R.matrix[i*N+j]+=a[m]*Q.matrix[m*N+i];
}
for(int m=0;m<N;++m)
{
b[m]-=R.matrix[i*N+j]*Q.matrix[m*N+i];
}
}
double norm=0;
for(int i=0;i<N;++i)
{
norm+=b[i]*b[i];
}
norm=(double)sqrt(norm);
R.matrix[j*N+j]=norm;
for (int i=0;i<N;++i)
{
Q.matrix[i*N+j]=b[i]/norm;
}
}
delete[]a;
delete[]b;
}
Matrix Matrix::diag()
{
if (row!=col)
{
Matrix m(0,0);
cout<<"矩阵不合法"<<endl;
return m;
}
Matrix m(row,row);
for(int i=0;i<row;i++)
{
m.matrix[i*row+i]=matrix[i*row+ i];
}
return m;
}
Matrix Matrix::eig_val(int _iters)
{
if (quantity==0||row!=col)
{
cout<<"矩阵为空或者非方阵！"<< endl;
Matrix rets(0,0);
return rets;
}
const int N=row;
Matrix matcopy(*this);//备份矩阵
Matrix Q(N,N),R(N,N);
for (int k=0;k<_iters;++k)
{
QR(Q,R);
*this=R*Q;
}
Matrix val=diag();
*this=matcopy;//恢复原始矩阵；
return val;
}
Matrix Matrix::eig_vect(int _iters)
{
if(quantity==0||row!=col)
{
cout<<"矩阵为空或者非方阵！"<<endl;
Matrix rets(0,0);
return rets;
}
if(determinant()==0)
{
cout <<"非满秩矩阵无法分解计算特征向量！"<<endl;
Matrix rets(0,0);
return rets;
}
Matrix matcopy(*this);
Matrix eigenValue=eig_val(_iters);
Matrix ret(row,row);
const int NUM=col;
double eValue;
double sum,midSum,diag;
Matrix copym(*this);
for(int count=0;count<NUM;++count)
{
*this=copym;
eValue=eigenValue[count][count];
for(int i=0;i<col;++i)//A-lambda*I
{
matrix[i*col+i]-=eValue;
}
for(int i=0;i<row-1;++i)
{
diag=matrix[i*col+i];
for(int j=i;j<col;++j)
{
matrix[i*col+j]/=diag;
}
for (int j=i+1;j<row;++j)
{
diag=matrix[j*col+ i];
for (int q=i;q<col;++q)
{
matrix[j*col+q]-=diag*matrix[i*col+q];
}
}
}
midSum=ret.matrix[(ret.row-1)*ret.col+count]=1;
for (int m=row-2;m>=0;--m)
{
sum=0;
for(int j=m+1;j<col;++j)
{
sum+=matrix[m*col+j]*ret.matrix[j*ret.col+count];
}
sum=-sum/matrix[m*col+m];
midSum+=sum*sum;
ret.matrix[m*ret.col+count]=sum;
}
midSum=sqrt(midSum);
for (int i=0;i<ret.row;++i)
{
ret.matrix[i*ret.col+count]/=midSum;
}
}
*this=matcopy;
return ret;
}
void menu()
{
cout<<"选择："<<endl;
cout<<"1.+"<<endl<<"2.-"<<endl<<"3.*"<<endl;
cout<<"4.求逆"<<endl<<"5.求行列式 \n6.矩阵的转置" <<endl;
cout<<"7.AX=b求解操作"<<endl;
cout<<"8.利用（）运算符重载可以改变某个矩阵元素的值\n";
cout<<"9.求矩阵的秩\n";
cout<<"请输入：";
int i;int n=3;
cin>>i;
Matrix a(n,n);
//cin>>a;
Matrix c(n,n);
switch(i) {
case 1:{
cout<<"请输入矩阵："<<endl;
cin>>a;
Matrix b(n,n);
cin>>b;
c=a+b;cout<<"a+b:\n"<<c;
break;
}
case 2:{
cout<<"请输入矩阵："<<endl;
cin>>a;
Matrix b(n,n);
cin>>b;
c=a-b;
cout<<"a-b:\n"<<c;
break;
}
case 3:{
cout<<"请输入矩阵："<<endl;
cin>>a;
Matrix b(n,n);
cin>>b;
c=a*b;cout<<"a*b:\n"<<c;
break;
}
case 4:{
cout<<"请输入矩阵："<<endl;
cin>>a;
c=a.inverse();
cout<<"矩阵a的逆\n"<<endl<<c<<endl;
break;
}
case 5:{
cout<<"请输入矩阵："<<endl;
cin>>a;
int det;
det=a.determinant();
cout<<"a的行列式:\n"<<det<<endl;
break;
}
case 6:{
cout<<"请输入矩阵："<<endl;
cin>>a;
c=a.transpose();
cout<<"实现矩阵的转置操作\n"<<c<<endl;
break;
}
case 7:{
cout<<"请输入矩阵："<<endl;
cin>>a;
Matrix b(n,n);
cin>>b;
c=solveAb(a,b);
cout<<"实现AX=b求解操作\n"<<endl;cout<<c;
break;
}
case 8:{
cout<<"请输入矩阵："<<endl;
cin>>a;
int i,j,n;
cout<<"利用（）运算符重载可以改变某个矩阵元素的值"<<endl;
cout<<"请输入row："<<endl; cin>>i;
cout<<"请输入col："<<endl; cin>>j;
cout<<"请输入n："<<endl; cin>>n;
a(i,j)=n; //利用（）运算符重载可以改变某个矩阵元素的值
break;
}
case 9:{
int r;
cout<<"请输入矩阵："<<endl;
cin>>a;
Matrix temp(n,n);
temp=a.gaussianEliminate();
cout<<"temp\n"<<temp;
r=a.rank(temp);
cout<<endl;
cout<<r<<endl;
break;
}
}
}
int main()
{
int choose=1;
menu();
while(choose)
{
cout<<"是否继续：1.继续；0.退出\n";
cin>>choose;
if(choose==1)
menu();
else
break;
}
return 0;
}