线性代数-矩阵-【1】矩阵汇总 C和C++的实现

矩阵的知识点之多足以写成一本线性代数。

在C++中，我们把矩阵封装成类。。

程序清单：

Matrix.h//未完待续

#ifndef _MATRIX_H
#define _MATRIX_H

#include<iostream>
#include<vector>
using namespace std;

template <typename T>
class Matrix
{
public://矩阵基本运算
    Matrix operator*(const Matrix<T> &matrix);                //运算符重载“*”重载为点乘
    Matrix operator+(const Matrix<T> &matrix);                //运算符重载“+”为矩阵加法
    Matrix operator-(const Matrix<T> &matrix);                //运算符重载“-”为矩阵减法
    Matrix operator|(const Matrix<T> &matrix);                //重载运算符“|”为为矩阵增广（把b矩阵并在a矩阵后面）
    Matrix operator*(const double input);                    //重载运算符“*”常数乘法
    Matrix operator*(const float input);                    //同上
    Matrix operator*(const int input);                        //同上

    double determinant();                                    //求矩阵行列式的值
    double det_recursion(Matrix<T> &matrix);                //求矩阵行列式的值 递归
    Matrix<T> getAlgebraicCofactor(int row,int column);        //获取代数余子式
    bool calAdjugateMatrix(Matrix<double> *matrix);            //求伴随矩阵
    Matrix transpose();                                        //获取矩阵的转置
    Matrix reduceAnIdentity(int number);                    //产生一个秩为number的单位矩阵
    bool GaussianElimination();                                //高斯消元
    bool rref();                                            //化简矩阵（reduced row echelon form）行最简式
    bool rrefmovie();                                        //化简矩阵（reduced row echelon form）行最简式，并打印过程
    bool inverse();                                            //矩阵的逆
    bool clear();                                            //清空矩阵
public://矩阵增删补改
    Matrix();
    Matrix(int rows,int columns);                            //产生一个行数为rows,列数为columnns的矩阵
    virtual ~Matrix();
    int getRows()const;                                                //获取行数
    int getColumns()const;                                            //获取列数
    bool getSpecifiedElem(int row,int column, T *elem)const;        //获取第i行第j个元素 
    bool getSpecifiedRow(int index,vector<T> *vec)const;            //获取第index行元素
    bool getSpecifiedColumn(int index,vector<T> *vec)const;            //获取第index列元素
    bool setSpecifiedElem(int row,int column,T value);                //设置第row行第column个元素的值为value
    bool addOneRowToBack(vector<T> &vec);                            //往最底插入一行
    bool addOneColumToBack(vector<T> &vec);                            //往最后插入一列
    bool exchangeRows(int index1st,int index2nd);                    //交换矩阵的两行
    bool relevantCheck(vector<T> &vecA,vector<T>  &vecB);            //两个向量的相关性检查
    bool invertibleCheck();                                            //可逆性检查
    void printfAll();                                                //打印所有元素
private:
    int m_iRows;
    int m_iColumns;
    vector<vector<T>>  m_vecMatrix;
};

template <typename T>
Matrix<T>::Matrix()
{
    m_iRows = 0;
    m_iColumns =0;
    m_vecMatrix.clear();
}

template <typename T>
Matrix<T>::Matrix(int rows,int columns)
{
    vector<T> tempVec;    

    m_iRows = 0;
    m_iColumns =0;
    m_vecMatrix.clear();

    //reduce a vector for adding
    for(int i=0;i<columns;i++)
    {
        tempVec.push_back(0);
    }

    //puduce a zero matirx
    for(int i=0;i<rows;i++)
    {
        addOneRowToBack(tempVec);
    }
}

template <typename T>
Matrix<T>::~Matrix()
{
    m_vecMatrix.clear();
}

template <typename T>
int Matrix<T>::getRows()const
{
    return m_iRows;
}

template <typename T>
int Matrix<T>::getColumns()const
{
    return m_iColumns;
}

template <typename T>
bool Matrix<T>::getSpecifiedElem(int row,int column, T *elem)const
{
    if(row>=m_iRows || column>=m_iColumns)
    {
        return false;
    }

    *elem = this->m_vecMatrix[row][column];
    return true;
}

template <typename T>
bool Matrix<T>::getSpecifiedRow(int index,vector<T> *vec)const
{
    if(index > m_iRows)
    {
        return false;
    }

    vec->clear();
    for(int i=0;i<m_iColumns;i++)
    {
        vec->push_back(m_vecMatrix[index][i]);
    }

    return true;
}

template <typename T>
bool Matrix<T>::getSpecifiedColumn(int index,vector<T> *vec)const//获取第index列元素
{
    if(index > m_iColumns)
    {
        return false;
    }

    vec->clear();
    for(int i=0;i<m_iRows;i++)
    {
        vec->push_back(m_vecMatrix[i][index]);
    }

    return true;    
}

template <typename T>
bool Matrix<T>::setSpecifiedElem(int row,int column,T value)    //设置第row行第column个元素的值为value
{
    if(row > m_iRows-1 ||column > m_iColumns-1)
    {
        return false;
    }    

    m_vecMatrix[row][column]=value;
    return true;
}

template <typename T>
bool Matrix<T>::addOneRowToBack(vector<T> &vec)
{
    /*Matrix has had datas*/
    if(m_iColumns !=0)
    {
        if(vec.size() != m_iColumns)//input data columns not equal matrix columns
        {
            cout<<"addOneRowToBack(): columns not equal"<<endl;
            return false ;
        }
    }

    /*Adding vec to m_vecDatas*/
    m_vecMatrix.push_back(vec);
    m_iRows ++;
    m_iColumns = vec.size();
    return true;    
}

template <typename T>
bool Matrix<T>::addOneColumToBack(vector<T> &vec)
{
    /*Matrix has had datas*/
    if(m_iRows !=0)
    {
        if(vec.size() != m_iRows)//input data columns not equal matrix columns
        {
            cout<<"addOneColumsToBack(): rows not equal"<<endl;
            return false ;
        }        
    }
    else 
    {
        vector<T> tempVec;
        m_iRows = vec.size();
        for(int i=0;i<m_iRows;i++)
        {
            m_vecMatrix.push_back(tempVec); 
        }
    }

    /*Adding vec to m_vecDatas*/ 
    for(int i=0;i<m_iRows;i++)
    {
        m_vecMatrix[i].push_back(vec[i]);
    }
    m_iColumns ++;
    m_iRows = vec.size();
    return true;    
}

template <typename T>
bool Matrix<T>::exchangeRows(int index1st,int index2nd)
{
    /*input leagality check*/
    if(index1st >= m_iRows || index2nd >= m_iRows)
    {
        cout<<"exchangeRows():index illeagel";
        return false;
    }

    /*Exchange*/
    Matrix<T> outputMatrix = *this;
    vector<T> tempVec = outputMatrix.m_vecMatrix[index1st];
    outputMatrix.m_vecMatrix[index1st] = outputMatrix.m_vecMatrix[index2nd]; 
    outputMatrix.m_vecMatrix[index2nd] = tempVec;
    *this = outputMatrix;
    return true;
}

template <typename T>
Matrix<T> Matrix<T>::reduceAnIdentity(int number)
{
    Matrix<T> outputMatrix(number,number);    

    if(number <= 0)
    {
        return outputMatrix;
    }

    //change matrixdata[i][i] to 1
    for(int i=0;i<number;i++)
    {
        outputMatrix.m_vecMatrix[i][i] = (T)1;
    }

    return outputMatrix;
}

template <typename T>
bool Matrix<T>::GaussianElimination()
{
    Matrix<T> outputMatrix = *this;

    /*Gaussian elmiation*/
    for(int k=0;k<outputMatrix.m_iRows;k++)
    {
        /*if all the pivot have been found*/
        if(k>=m_iColumns)
        {
            break;
        }

        /*exchange rows downward to find the row's pivot*/
        for(int i=k+1;i<outputMatrix.m_iRows;i++)
        {
            /*pivot is non-zero*/
            if(outputMatrix.m_vecMatrix[k][k] != 0)
            {
                //T temp = outputMatrix.m_vecMatrix[0][0];
                break;
            }
            else
            {            
                if(i < outputMatrix.m_iRows)
                {
                    outputMatrix.exchangeRows(k,i);
                }
            }
        }

        /*if there is no pivot in this row*/
        if(outputMatrix.m_vecMatrix[k][k] == 0)
        {
            break;
        }

        /*elimination:The rows below pivot row subtract times of pivot row*/
        for(int i=k+1;i<outputMatrix.m_iRows;i++)
        {
            double RowsfirstData = outputMatrix.m_vecMatrix[i][k]/outputMatrix.m_vecMatrix[k][k];//Save the first data of next(k+1) rows 
            if(RowsfirstData != 0) 
            {
                outputMatrix.m_vecMatrix[i][k]=0;
                for(int j=k+1;j<outputMatrix.m_iColumns;j++)
                {
                    outputMatrix.m_vecMatrix[i][j] -= RowsfirstData*outputMatrix.m_vecMatrix[k][j] ;
                }
            }
        }
    }

    *this = outputMatrix;
    return true;
}

template <typename T>
bool Matrix<T>::rref()
{
    Matrix<T> outputMatrix = *this;
    int rank=0;//the rank of the matrix, how many columns's pivot will it has(-1)

    /*Gaussian elmiation*/
    for(int k=0;k<outputMatrix.m_iRows;k++)
    {
        /*if all the pivot elem have been found*/
        if(k>=m_iColumns)
        {
            break;
        }

        /*exchange rows downward to find the pivot row*/
        for(int i=k+1;i<outputMatrix.m_iRows;i++)
        {
            /*pivot is non-zero*/
            if(outputMatrix.m_vecMatrix[k][k] != 0)
            {
                //T temp = outputMatrix.m_vecMatrix[0][0];
                rank++;
                break;
            }
            else
            {            
                if(i < outputMatrix.m_iRows)
                {
                    outputMatrix.exchangeRows(k,i);
                }
            }
        }

        /*if there is no pivot in this row*/
        if(outputMatrix.m_vecMatrix[k][k] == 0)
        {
            break;
        }

        /*elimination:The rows below pivot row subtract times of pivot row*/
        for(int i=k+1;i<outputMatrix.m_iRows;i++)
        {
            double RowsfirstData = outputMatrix.m_vecMatrix[i][k]/outputMatrix.m_vecMatrix[k][k];//Save the first data of next(k+1) rows 
            if(RowsfirstData != 0) 
            {
                outputMatrix.m_vecMatrix[i][k]=0;
                for(int j=k+1;j<outputMatrix.m_iColumns;j++)
                {
                    outputMatrix.m_vecMatrix[i][j] -= RowsfirstData*outputMatrix.m_vecMatrix[k][j] ;
                }
            }
        }
    }

    /*normalizing:set all pivots to 1*/
    for(int i=0;i<outputMatrix.m_iRows;i++)
    {
        for(int j=0;j<outputMatrix.m_iColumns;j++)
        {
            if(outputMatrix.m_vecMatrix[i][j] !=0 )//pivot has been foound
            {
                double pivot = outputMatrix.m_vecMatrix[i][j];//get pivot
                for(int k=i;k<outputMatrix.m_iColumns;k++)
                {
                    outputMatrix.m_vecMatrix[i][k] /=pivot;
                }
                break;
            }
        }
    }

    /*Back substitution*/
    for(int i = rank;i>=1;i--)
    {
        /*find a first non-zero elem (It is pivot)*/
        for(int j=0;j<outputMatrix.m_iColumns;j++)
        {
            double times=0;
            if(outputMatrix.m_vecMatrix[i][j] !=0)//pivot found
            {
                for(int l=i-1;l>=0;l--)
                {
                    times = outputMatrix.m_vecMatrix[l][j]/outputMatrix.m_vecMatrix[i][j]; 
                    for(int k=j;k<outputMatrix.m_iColumns;k++)//tims of this row subtract by each columns in upon row 
                    {
                        outputMatrix.m_vecMatrix[l][k] -= times*outputMatrix.m_vecMatrix[i][k];
                    }
                }
                break;
            }
        }
    }

    *this = outputMatrix;
    return true;
}

template <typename T>
bool Matrix<T>::rrefmovie()
{
    Matrix<T> outputMatrix = *this;
    int rank=0;//the rank of the matrix, how many columns's pivot will it has(-1)

    /*Gauss elmiation*/
    cout<<"Gauss elimination:"<<endl;
    outputMatrix.printfAll();
    for(int k=0;k<outputMatrix.m_iRows;k++)
    {
        /*If all the pivot elem have been found*/
        if(k>=m_iColumns)
        {
            break;
        }

        /*Exchange rows downward to find the pivot row*/
        for(int i=k+1;i<outputMatrix.m_iRows;i++)
        {
            /*Pivot is non-zero*/
            if(outputMatrix.m_vecMatrix[k][k] != 0)
            {
                rank++;
                break;
            }
            else
            {            
                if(i < outputMatrix.m_iRows)
                {
                    outputMatrix.exchangeRows(k,i);
                }
            }
            if(k!=i)
            {
                cout<<"row"<<k+1<<" exchange row"<<i<<endl;//Debug
                outputMatrix.printfAll();
            }
        }

        /*If there is no pivot in this row*/
        if(outputMatrix.m_vecMatrix[k][k] == 0)
        {
            break;
        }

        /*Elimination:The rows below pivot row subtract times of pivot row*/
        for(int i=k+1;i<outputMatrix.m_iRows;i++)
        {
            double RowsfirstData = outputMatrix.m_vecMatrix[i][k]/outputMatrix.m_vecMatrix[k][k];//Save the first data of next(k+1) rows 
            if(RowsfirstData != 0) 
            {
                outputMatrix.m_vecMatrix[i][k]=0;
                for(int j=k+1;j<outputMatrix.m_iColumns;j++)
                {
                    outputMatrix.m_vecMatrix[i][j] -= RowsfirstData*outputMatrix.m_vecMatrix[k][j] ;
                }
            }
            cout<<"row"<<i+1<<" - "<<RowsfirstData<<"*"<<"row"<<k+1<<endl;//Debug
            outputMatrix.printfAll();
        }
    }

    /*Normalizing:set all rows pivot to 1*/
    for(int i=0;i<outputMatrix.m_iRows;i++)
    {
        for(int j=0;j<outputMatrix.m_iColumns;j++)
        {
            if(outputMatrix.m_vecMatrix[i][j] !=0 )//pivot has been foound
            {
                double pivot = outputMatrix.m_vecMatrix[i][j];//get pivot
                for(int k=i;k<outputMatrix.m_iColumns;k++)
                {
                    outputMatrix.m_vecMatrix[i][k] /=pivot;
                }
                cout<<"row"<<i+1<<" / "<<pivot<<endl;//Debug
                outputMatrix.printfAll();//Debug
                break;
            }
        }
    }


    /*Back substitution*/
    cout<<"Back substitution:"<<endl;
    for(int i = rank;i>=1;i--)
    {
        /*find a first non-zero elem (It is pivot)*/
        for(int j=0;j<outputMatrix.m_iColumns;j++)
        {
            double times=0;
            if(outputMatrix.m_vecMatrix[i][j] !=0)//pivot found
            {
                for(int l=i-1;l>=0;l--)
                {
                    times = outputMatrix.m_vecMatrix[l][j]/outputMatrix.m_vecMatrix[i][j]; 
                    for(int k=j;k<outputMatrix.m_iColumns;k++)//tims of this row subtract by each columns in upon row 
                    {
                        outputMatrix.m_vecMatrix[l][k] -= times*outputMatrix.m_vecMatrix[i][k];
                    }
                    cout<<"row"<<l+1<<" - "<<times<<"*"<<"row"<<i+1<<endl;
                    outputMatrix.printfAll();
                }
                break;
            }
        }
    }

    *this = outputMatrix;
    return true;
}

template <typename T>
void Matrix<T>::printfAll()
{
    for(int j=0;j<m_iRows;j++)
    {
        cout<<"| ";
        for(int i=0;i<m_iColumns;i++)
        {
            cout<<m_vecMatrix[j][i]<<" ";
        }
        cout<<"|"<<endl;
    }
    cout<<" 
"<<endl;
}

template <typename T>
Matrix<T> Matrix<T>::operator+(const Matrix<T> &matrix)                //运算符重载“+”为矩阵加法
{
    /*matrix leagality check*/
    if(this->m_iRows != matrix.getRows() || this->m_iColumns != matrix.getColumns())
    {
        return *this;
    }

    Matrix<T> outputMatrix;
    vector<T> tempVec;
    for(int i=0;i<this->m_iRows;i++)
    {
        tempVec.clear();
        for(int j=0;j<this->m_iColumns;j++)
        {
            tempVec.push_back(this->m_vecMatrix[i][j] + matrix.m_vecMatrix[i][j]);
        }
        outputMatrix.addOneRowToBack(tempVec);
    }

    return outputMatrix;
}

template <typename T>
Matrix<T> Matrix<T>::operator-(const Matrix<T> &matrix)                //运算符重载“-”为矩阵减法
{
    /*matrix leagality check*/
    if(this->m_iRows != matrix.getRows() || this->m_iColumns != matrix.getColumns())
    {
        return *this;
    }

    Matrix<T> outputMatrix;
    vector<T> tempVec;
    for(int i=0;i<this->m_iRows;i++)
    {
        tempVec.clear();
        for(int j=0;j<this->m_iColumns;j++)
        {
            tempVec.push_back(this->m_vecMatrix[i][j] - matrix.m_vecMatrix[i][j]);
        }
        outputMatrix.addOneRowToBack(tempVec);
    }

    return outputMatrix;    
}

template <typename T>
Matrix<T> Matrix<T>::operator|(const Matrix<T> &matrix)
{
    /*input size check*/
    if(this->m_iRows != matrix.getRows() )
    {
        return *this;
    }

    /**/
    Matrix<T> outputMatrix = *this;
    for(int i=0;i<m_iColumns;i++)
    {
        vector<T> tempVec ;
        if(!matrix.getSpecifiedColumn(i,&tempVec))
        {
            return outputMatrix;
        }
        outputMatrix.addOneColumToBack(tempVec);
    }

    return outputMatrix; 
}


template <typename T>
Matrix<T> Matrix<T>::operator*(const Matrix<T> &matrix)                    //运算符重载*重载为点乘
{
    /*matrix leagality check*/
    if(this->m_iColumns  !=  matrix.getRows())
    {
        cout<<"operator*():input ileagal"<<endl;
        return *this;
    }


    /*Caculate point multiply*/
    //function 1st
    Matrix<T> outputMatrix;
    vector<T> tempVec;
    T tempData;
    for(int j=0;j<m_iRows;j++)
    {
        for(int k=0;k<matrix.m_iColumns;k++)
        {
            tempData =0;
            for(int i=0;i<m_iColumns;i++)
            {
                tempData += this->m_vecMatrix[j][i] * matrix.m_vecMatrix[i][k];
            }
            tempVec.push_back(tempData);
        }
        outputMatrix.addOneRowToBack(tempVec);
        tempVec.clear(); //clear for next rows adding
    }

    //function 2nd
    //vector<T> tempVecRow;
    //vector<T> tempVecColumn;
    //vector<T> tempVecResult;
    //int temp;

    //for(int k=0;k<m_iRows;k++)
    //{
    //    for(int i=0;i<matrix.getColumns();i++)
    //    {
    //        this->getSpecifiedRow(k,&tempVecRow);//get multiplier row
    //        matrix.getSpecifiedColumn(i,&tempVecColumn);//get multiplicand column
    //        temp=0;
    //        for(int j=0;j<(int)tempVecRow.size();j++)
    //        {
    //            temp += tempVecRow[j]*tempVecColumn[j];
    //        }
    //        tempVecResult.push_back(temp);
    //    }
    //    outputMatrix.addOneRowToBack(tempVecResult);
    //    tempVecResult.clear(); //clear for next rows adding
    //}

    return outputMatrix;
}

/*常数乘法*/
template <typename T>
Matrix<T> Matrix<T>::operator*(const double input)                        //重载运算符“*”常数乘法
{
    Matrix<T> outputMatrix = *this;
    for(int i=0;i<m_iRows;i++)
    {
        for(int j=0;j<m_iColumns;j++)
        {
            outputMatrix.m_vecMatrix[i][j] *= input;
        }
    }
    return outputMatrix;
}
template <typename T>
Matrix<T> Matrix<T>::operator*(const float input)
{
    Matrix<T> outputMatrix = *this;
    for(int i=0;i<m_iRows;i++)
    {
        for(int j=0;j<m_iColumns;j++)
        {
            outputMatrix.m_vecMatrix[i][j] *= input;
        }
    }
    return outputMatrix;
}
template <typename T>
Matrix<T> Matrix<T>::operator*(const int input)
{
    Matrix<T> outputMatrix = *this;
    for(int i=0;i<m_iRows;i++)
    {
        for(int j=0;j<m_iColumns;j++)
        {
            outputMatrix.m_vecMatrix[i][j] *= input;
        }
    }
    return outputMatrix;
}

template <typename T>
double Matrix<T>::determinant()
{
    //not square
    if(m_iColumns != m_iRows)
        return 0;

    /*function 1st:recursion*/
    //double result = det_recursion(*this);

    /*function 2nd*/
    //1:Gaussian elimination 
    //2:Diagonal multipication
    Matrix<T> tempMatrix = *this;
    tempMatrix.GaussianElimination();

    double result=1;
    for(int i=0;i<m_iRows;i++)
    {
        result*=tempMatrix.m_vecMatrix[i][i];
    }

    return result;
}


template <typename T>
double Matrix<T>::det_recursion(Matrix<T> &matrix)
{
    double result=0;
    if(matrix.m_iRows!=2)
    {
        for(int i=0;i<matrix.m_iColumns;i++)
        {
            Matrix<T> tempMatrix = matrix.getAlgebraicCofactor(i,0); 
            if(i%2==0)//even
            {
                result+=matrix.m_vecMatrix[i][0]*det_recursion(tempMatrix);
            }
            else//old
            {
                result-=matrix.m_vecMatrix[i][0]*det_recursion(tempMatrix);
            }
        }
        return result;
    }
    else
    {
        return (double)matrix.m_vecMatrix[0][0]*matrix.m_vecMatrix[1][1] - (double)matrix.m_vecMatrix[0][1]*matrix.m_vecMatrix[1][0];
    }
}


template <typename T>
Matrix<T> Matrix<T>::getAlgebraicCofactor(int row,int column)
{
    if(row<0 || column<0 || row >m_iColumns-1 || column>m_iColumns-1)
    {
        return *this;    
    }

    Matrix<T> outputMatrix(this->m_iRows-1,this->m_iColumns-1);

    for(int i=0;i<this->m_iRows;i++)
    {
        for(int j=0;j<this->m_iColumns;j++)
        {
            if(i<row )
            {
                if(j<column)
                    outputMatrix.m_vecMatrix[i][j] = this->m_vecMatrix[i][j];
                else if(j>column)
                    outputMatrix.m_vecMatrix[i][j-1] = this->m_vecMatrix[i][j];
            }
            else if(i>row)
            {
                if(j<column)
                    outputMatrix.m_vecMatrix[i-1][j] = this->m_vecMatrix[i][j];
                else if(j>column)
                    outputMatrix.m_vecMatrix[i-1][j-1] = this->m_vecMatrix[i][j];
            }

        }
    }
    return outputMatrix;
}

template <typename T>
bool Matrix<T>::calAdjugateMatrix(Matrix<double> *matrix)
{
    if(determinant() ==0 )
        return false ;

    Matrix<double> outputMatrix(m_iRows,m_iColumns);
    for(int i=0;i<m_iRows;i++)
    {
        for(int j=0;j<m_iColumns;j++)
        {
            outputMatrix.m_vecMatrix[i][j] = getAlgebraicCofactor(i,j).determinant();
        }
    }
    *matrix = outputMatrix;
    return true;
}

template <typename T>
Matrix<T> Matrix<T>::transpose()
{
    Matrix<T> tempMatrix;
    vector<T> tempVec;

    /*get transpose*/
    for(int i=0;i<m_iColumns;i++)
    {
        this->getSpecifiedRow(i,&tempVec);
        tempMatrix.addOneColumToBack(tempVec);
    }

    /*swap rows number and columns number*/
    m_vecMatrix = tempMatrix.m_vecMatrix;
    int temp = m_iColumns;
    m_iColumns = m_iRows;
    m_iRows = temp;

    return tempMatrix;
}

template <typename T>
bool Matrix<T>::relevantCheck(vector<T> &vecA,vector<T>  &vecB)
{
    if(vecA.size()!=vecB.size())
    {
        return false;
    }

    int length = vecA.size();
    T timesofA,timesofB;
    int i;
    for(i=0;i<length;i++)
    {
        if(vecA[i]!=0 || vecB[i]!=0)
        {
            timesofA=vecA[i];
            timesofB=vecB[i];
            break;
        }
        if(i == length-1)
        {
            return true;
        }
    }
    i+=1;
    for(i;i<length;i++)
    {
        if(timesofA*vecB[i] != timesofB*vecA[i])
            return false;
    }
    return true;

}

template <typename T>
bool Matrix<T>::invertibleCheck()
{

    /*rows relevant check*/
    for(int i=0;i<m_iRows-1;i++)
    {
        vector<T> vecA;
        vector<T> vecB;
        for(int j=i+1;j<m_iRows;j++)
        {
            getSpecifiedRow(i,&vecA);
            getSpecifiedRow(j,&vecB);
            if(relevantCheck(vecA,vecB))
            {
                return false;
            }
        }
    }

    /*columns relevant check*/
    for(int i=0;i<m_iRows-1;i++)
    {
        vector<T> vecA;
        vector<T> vecB;
        for(int j=i+1;j<m_iRows;j++)
        {
            getSpecifiedColumn(i,&vecA);
            getSpecifiedColumn(j,&vecB);
            if(relevantCheck(vecA,vecB))
            {
                return false;
            }
        }
    }

    return true;
}


template <typename T>
bool Matrix<T>::inverse()
{
    /*Determinant equart 0,there is no inverse.*/
    if(this->determinant() ==0 )
    {
        return false;
    }

    /*Make an Identity*/
    Matrix<T> I =this->reduceAnIdentity( this->m_iRows );

    /*Augmented by I and do rref*/
    Matrix<T> augmentationMatrix=*this|I;
    augmentationMatrix.rref();
    //augmentationMatrix.rrefmovie();

    Matrix<T> outputMatrix;
    vector<T> tempVec;
    for(int i=0;i<m_iColumns;i++)
    {
        augmentationMatrix.getSpecifiedColumn(m_iRows+i,&tempVec);
        outputMatrix.addOneColumToBack(tempVec);
    }

    *this  = outputMatrix;

    return true;
}

template <typename T>
bool Matrix<T>::clear()
{
    m_vecMatrix.clear();
    return true;
}
#endif

分组解析：

基本成员解析：

• 矩阵 -【2】 矩阵生成：http://www.cnblogs.com/HongYi-Liang/p/7275278.html

算术运算：

• 矩阵 - 【3】矩阵加减：http://www.cnblogs.com/HongYi-Liang/p/7287403.html
• 矩阵 - 【4】矩阵点乘：http://www.cnblogs.com/HongYi-Liang/p/7287324.html
• 矩阵 - 叉乘：

其他运算：

• 矩阵 - 矩阵转置：http://www.cnblogs.com/HongYi-Liang/p/7287495.html
• 矩阵 - 【5】矩阵化简：http://www.cnblogs.com/HongYi-Liang/p/7464850.html
• 矩阵 - 行列式：
• 矩阵 - 逆矩阵：