在图像处理中经常用矩阵，抽时间看了矩阵的内容，还是有很多数学应用和解决方案的，整理了一下。
1.行主映射，列主映射
在二维数组中，可以把二维映射到一维。
行主映射：从第一行开始，一次对每行引索从左至右编号，把二维数组映射为[0,n-1].
列主映射：从第一列开始，一次对每列引索从左至右编号
如：
2.矩阵中的转置，相加（两个矩阵行数和列数分别相等），相乘（a的列数等于b的行数），运算符重载
例：matrix.h
#ifndef matrix_ #define matrix_ #include "myExceptions.h" using namespace std; template<class T> class matrix { friend ostream& operator<<(ostream&, const matrix<T>&); public: matrix(int theRows = 0, int theColumns = 0); matrix(const matrix<T>&); ~matrix() {delete [] element;} int rows() const {return theRows;} int columns() const {return theColumns;} T& operator()(int i, int j) const; matrix<T>& operator=(const matrix<T>&); matrix<T> operator+() const; // unary + matrix<T> operator+(const matrix<T>&) const; matrix<T> operator-() const; // unary minus matrix<T> operator-(const matrix<T>&) const; matrix<T> operator*(const matrix<T>&) const; matrix<T>& operator+=(const T&); private: int theRows, // number of rows in matrix theColumns; // number of columns in matrix T *element; // element array }; template<class T> matrix<T>::matrix(int theRows, int theColumns) {// matrix constructor. // validate theRows and theColumns if (theRows < 0 || theColumns < 0) throw illegalParameterValue("Rows and columns must be >= 0"); if ((theRows == 0 || theColumns == 0) && (theRows != 0 || theColumns != 0)) throw illegalParameterValue ("Either both or neither rows and columns should be zero"); // create the matrix this->theRows = theRows; this->theColumns = theColumns; element = new T [theRows * theColumns]; } template<class T> matrix<T>::matrix(const matrix<T>& m) {// Copy constructor for matrices. // create matrix theRows = m.theRows; theColumns = m.theColumns; element = new T [theRows * theColumns]; // copy each element of m copy(m.element, m.element + theRows * theColumns, element); } template<class T> matrix<T>& matrix<T>::operator=(const matrix<T>& m) {// Assignment. (*this) = m. if (this != &m) {// not copying to self delete [] element; theRows = m.theRows; theColumns = m.theColumns; element = new T [theRows * theColumns]; // copy each element copy(m.element, m.element + theRows * theColumns, element); } return *this; } template<class T> T& matrix<T>::operator()(int i, int j) const {// Return a reference to element (i,j). if (i < 1 || i > theRows || j < 1 || j > theColumns) throw matrixIndexOutOfBounds(); return element[(i - 1) * theColumns + j - 1]; } template<class T> matrix<T> matrix<T>::operator+(const matrix<T>& m) const {// Return w = (*this) + m. if (theRows != m.theRows || theColumns != m.theColumns) throw matrixSizeMismatch(); // create result matrix w matrix<T> w(theRows, theColumns); for (int i = 0; i < theRows * theColumns; i++) w.element[i] = element[i] + m.element[i]; return w; } template<class T> matrix<T> matrix<T>:: operator-(const matrix<T>& m) const {// Return (*this) - m. if (theRows != m.theRows || theColumns != m.theColumns) throw matrixSizeMismatch(); // create result matrix w matrix<T> w(theRows, theColumns); for (int i = 0; i < theRows * theColumns; i++) w.element[i] = element[i] - m.element[i]; return w; } template<class T> matrix<T> matrix<T>::operator-() const {// Return w = -(*this). // create result matrix w matrix<T> w(theRows, theColumns); for (int i = 0; i < theRows * theColumns; i++) w.element[i] = -element[i]; return w; } template<class T> matrix<T> matrix<T>::operator*(const matrix<T>& m) const {// matrix multiply. Return w = (*this) * m. if (theColumns != m.theRows) throw matrixSizeMismatch(); matrix<T> w(theRows, m.theColumns); // result matrix // define cursors for *this, m, and w // and initialize to location of (1,1) element int ct = 0, cm = 0, cw = 0; // compute w(i,j) for all i and j for (int i = 1; i <= theRows; i++) {// compute row i of result for (int j = 1; j <= m.theColumns; j++) { // compute first term of w(i,j) T sum = element[ct] * m.element[cm]; // add in remaining terms for (int k = 2; k <= theColumns; k++) { ct++; // next term in row i of *this cm += m.theColumns; // next in column j of m sum += element[ct] * m.element[cm]; } w.element[cw++] = sum; // save w(i,j) // reset to start of row and next column ct -= theColumns - 1; cm = j; } // reset to start of next row and first column ct += theColumns; cm = 0; } return w; } template<class T> matrix<T>& matrix<T>::operator+=(const T& x) {// Increment all elements of *this by x. for (int i = 0; i < theRows * theColumns; i++) element[i] += x; return *this; } template<class T> ostream& operator<<(ostream& out, const matrix<T>& m) {// Put matrix m into the stream out. // One row per line. int k = 0; // index into element array for (int i = 0; i < m.theRows; i++) {// do row i for (int j = 0; j < m.theColumns; j++) out << m.element[k++] << " "; // row i finished out << endl; } return out; } // for some reason compiler can't create this on its own ostream& operator<<(ostream& out, const matrix<int>& m) {// Put matrix m into the stream out. // One row per line. int k = 0; // index into element array for (int i = 0; i < m.theRows; i++) {// do row i for (int j = 0; j < m.theColumns; j++) out << m.element[k++] << " "; // row i finished out << endl; } return out; } #endif matrix.cpp // test matrix class #include <iostream> #include "matrix.h" using namespace std; int main(void) { try { matrix<int> x(3,2), y, z; int i, j; for (i = 1; i <= 3; i++) for (j = 1; j <= 2; j++) x(i,j) = 2*i + j; cout << "Initialized x(i,j) = 2*i + j" << endl; cout << "x(3,1) = " << x(3,1) << endl; cout << "x is" << endl;; cout << x << endl; y = x; cout << "Assigned y = x" << endl; cout << "y is" << endl; cout << y << endl; x += 2; cout << "x incremented by 2 is" << endl; cout << x << endl; z = y + x; cout << "y + x is" << endl; cout << z << endl; cout << "-(y + x) is" << endl; cout << -z << endl; matrix<int> w(2,3); for (i = 1; i <= 2; i++) for (j = 1; j <= 3; j++) w(i,j) = i + j; cout << "Initialized w(i,j) = i + j" << endl; cout << "w is" << endl; cout << w << endl; z = y * w; cout << "y * w is" << endl; cout << z << endl; } catch (...) { cerr << "An exception has occurred" << endl;} return 0; } 3.特殊矩阵中又有：对角矩阵，三对角矩阵，下三角矩阵，上三角矩阵，对称矩阵等4稀疏矩阵：
在m*n 矩阵中，如果大多数元素都是0，就称为稀疏矩阵
而稀疏矩阵可以映射到一个线性表中，如果按行排列：
下面是稀疏矩阵重载操作符，矩阵转置，add：
sparseMatrix.h
// sparse matrix using an extended array list #ifndef sparseMatrix_ #define sparseMatrix_ #include <iostream> #include "matrixTerm.h" #include "extendedArrayList.h" #include "myExceptions.h" template<class T> class sparseMatrix { friend ostream& operator<< (ostream&, sparseMatrix<T>&); friend istream& operator>> (istream&, sparseMatrix<T>&); public: void transpose(sparseMatrix<T> &b); void add(sparseMatrix<T> &b, sparseMatrix<T> &c); private: int rows, // number of rows in matrix cols; // number of columns in matrix arrayList<matrixTerm<T> > terms; // list of nonzero terms }; // overload << template <class T> ostream& operator<<(ostream& out, sparseMatrix<T>& x) {// Put x in output stream. // put matrix characteristics out << "rows = " << x.rows << " columns = " << x.cols << endl; out << "nonzero terms = " << x.terms.size() << endl; // put terms, one per line for (arrayList<matrixTerm<T> >::iterator i = x.terms.begin(); i != x.terms.end(); i++) out << "a(" << (*i).row << ',' << (*i).col << ") = " << (*i).value << endl; return out; } // overload >> template<class T> istream& operator>>(istream& in, sparseMatrix<T>& x) {// Input a sparse matrix. // input matrix characteristics int numberOfTerms; cout << "Enter number of rows, columns, and #terms" << endl; in >> x.rows >> x.cols >> numberOfTerms; // set size of x.terms and ensure enough capacity x.terms.reSet(numberOfTerms); // input terms matrixTerm<T> mTerm; for (int i = 0; i < numberOfTerms; i++) { cout << "Enter row, column, and value of term " << (i + 1) << endl; in >> mTerm.row >> mTerm.col >> mTerm.value; x.terms.set(i, mTerm); } return in; } /****************************************************************/ // explict code tooverload with T = int for test as compiler // unable to generate // overload << ostream& operator<<(ostream& out, sparseMatrix<int>& x) {// Put x in output stream. // put matrix characteristics out << "rows = " << x.rows << " columns = " << x.cols << endl; out << "nonzero terms = " << x.terms.size() << endl; // put terms, one per line for (arrayList<matrixTerm<int> >::iterator i = x.terms.begin(); i != x.terms.end(); i++) out << "a(" << (*i).row << ',' << (*i).col << ") = " << (*i).value << endl; return out; } // overload >> istream& operator>>(istream& in, sparseMatrix<int>& x) {// Input a sparse matrix. // input matrix characteristics int numberOfTerms; cout << "Enter number of rows, columns, and #terms" << endl; in >> x.rows >> x.cols >> numberOfTerms; // set size of x.terms and ensure enough capacity x.terms.reSet(numberOfTerms); // input terms matrixTerm<int> mTerm; for (int i = 0; i < numberOfTerms; i++) { cout << "Enter row, column, and value of term " << (i + 1) << endl; in >> mTerm.row >> mTerm.col >> mTerm.value; x.terms.set(i, mTerm); } return in; } /****************************************************************/ template<class T> void sparseMatrix<T>::transpose(sparseMatrix<T> &b) {// Return transpose of *this in b. // set transpose characteristics b.cols = rows; b.rows = cols; b.terms.reSet(terms.size()); // initialize to compute transpose int* colSize = new int[cols + 1]; int* rowNext = new int[cols + 1]; // find number of entries in each column of *this for (int i = 1; i <= cols; i++) // initialize colSize[i] = 0; for (arrayList<matrixTerm<T> >::iterator i = terms.begin(); i != terms.end(); i++) colSize[(*i).col]++; // find the starting point of each row of b rowNext[1] = 0; for (int i = 2; i <= cols; i++) rowNext[i] = rowNext[i - 1] + colSize[i - 1]; // perform the transpose copying from *this to b matrixTerm<T> mTerm; for (arrayList<matrixTerm<T> >::iterator i = terms.begin(); i != terms.end(); i++) { int j = rowNext[(*i).col]++; // position in b mTerm.row = (*i).col; mTerm.col = (*i).row; mTerm.value = (*i).value; b.terms.set(j, mTerm); } } template<class T> void sparseMatrix<T>::add(sparseMatrix<T> &b, sparseMatrix<T> &c) {// Compute c = (*this) + b. // verify compatibility if (rows != b.rows || cols != b.cols) throw matrixSizeMismatch(); // incompatible matrices // set characteristics of result c c.rows = rows; c.cols = cols; c.terms.clear(); int cSize = 0; // define iterators for *this and b arrayList<matrixTerm<T> >::iterator it = terms.begin(); arrayList<matrixTerm<T> >::iterator ib = b.terms.begin(); arrayList<matrixTerm<T> >::iterator itEnd = terms.end(); arrayList<matrixTerm<T> >::iterator ibEnd = b.terms.end(); // move through *this and b adding like terms while (it != itEnd && ib != ibEnd) { // row-major index plus cols of each term int tIndex = (*it).row * cols + (*it).col; int bIndex = (*ib).row * cols + (*ib).col; if (tIndex < bIndex) {// b term comes later c.terms.insert(cSize++, *it); it++; } else {if (tIndex == bIndex) {// both in same position // append to c only if sum not zero if ((*it).value + (*ib).value != 0) { matrixTerm<T> mTerm; mTerm.row = (*it).row; mTerm.col = (*it).col; mTerm.value = (*it).value + (*ib).value; c.terms.insert(cSize++, mTerm); } it++; ib++; } else {// a term comes later c.terms.insert(cSize++, *ib); ib++; } } } // copy over remaining terms for (; it != itEnd; it++) c.terms.insert(cSize++, *it); for (; ib != ibEnd; ib++) c.terms.insert(cSize++, *ib); } #endif sparseMatrix.cpp: // test array based sparse matrix class #include <iostream> #include "sparseMatrix.h" int main(void) { sparseMatrix<int> a, b, c; // test input and output cin >> a; cout << "Matrix a is" << endl << a; cin >> b; cout << "Matrix b is" << endl << b; // test transpose a.transpose(c); cout << "The transpose of a is" << endl << c; // test add a.add(b,c); cout << "The sum of a and b is" << endl << c; return 0; }
