/// <summary>/// 計算矩陣的秩/// </summary>/// <param name="matrix">矩陣</param>/// <returns></returns>private static int Rank(double[][] matrix){  //matrix為空則直接默認已經是最簡形式  if (matrix == null || matrix.Length == 0) return 0;  //復制一個matrix到copy，之后因計算需要改動矩陣時并不改動matrix本身  double[][] copy = new double[matrix.Length][];  for (int i = 0; i < copy.Length; i++)  {    copy[i] = new double[matrix[i].Length];  }  for (int i = 0; i < matrix.Length; i++)  {    for (int j = 0; j < matrix[0].Length; j++)    {      copy[i][j] = matrix[i][j];    }  }  //先以最左側非零項的位置進行行排序  Operation1(copy);  //循環化簡矩陣  while (!isFinished(copy))  {    Operation2(copy);    Operation1(copy);  }  //過于趨近0的項，視作0，減小誤差  Operation3(copy);  //行最簡矩陣的秩即為所求  return Operation4(matrix);}/// <summary>/// 判斷矩陣是否變換到最簡形式（非零行數達到最少）/// </summary>/// <param name="matrix"></param>/// <returns>true:</returns>private static bool isFinished(double[][] matrix){  //統計每行第一個非零元素的出現位置  int[] counter = new int[matrix.Length];  for (int i = 0; i < matrix.Length; i++)  {    for (int j = 0; j < matrix[i].Length; j++)    {      if (matrix[i][j] == 0)      {        counter[i]++;      }      else break;    }  }  //后面行的非零元素出現位置必須在前面行的后面，全零行除外  for (int i = 1; i < counter.Length; i++)  {    if (counter[i] <= counter[i - 1] && counter[i] != matrix[0].Length)    {      return false;    }  }  return true;}/// <summary>/// 排序（按左側最前非零位位置自上而下升序排列）/// </summary>/// <param name="matrix">矩陣</param>private static void Operation1(double[][] matrix){  //統計每行第一個非零元素的出現位置  int[] counter = new int[matrix.Length];  for (int i = 0; i < matrix.Length; i++)  {    for (int j = 0; j < matrix[i].Length; j++)    {      if (matrix[i][j] == 0)      {        counter[i]++;      }      else break;     }  }  //按每行非零元素的出現位置升序排列  for (int i = 0; i < counter.Length; i++)  {    for (int j = i; j < counter.Length; j++)    {      if(counter[i]>counter[j])      {        double[] dTemp = matrix[i];        matrix[i] = matrix[j];        matrix[j] = dTemp;      }    }  }}/// <summary>/// 行初等變換（左側最前非零位位置最靠前的行，只保留一個）/// </summary>/// <param name="matrix">矩陣</param>private static void Operation2(double[][] matrix){  //統計每行第一個非零元素的出現位置  int[] counter = new int[matrix.Length];  for (int i = 0; i < matrix.Length; i++)  {    for (int j = 0; j < matrix[i].Length; j++)    {      if (matrix[i][j] == 0)      {        counter[i]++;      }      else break;    }  }  for (int i = 1; i < counter.Length; i++)  {    if (counter[i] == counter[i - 1] && counter[i] != matrix[0].Length)    {      double a = matrix[i - 1][counter[i - 1]];      double b = matrix[i][counter[i]]; //counter[i]==counter[i-1]      matrix[i][counter[i]] = 0;      for (int j = counter[i] + 1; j < matrix[i].Length; j++)      {        double c = matrix[i - 1][j];        matrix[i][j] -= (c * b / a);      }      break;    }  }}/// <summary>/// 將和0非常接近的數字視為0/// </summary>/// <param name="matrix"></param>private static void Operation3(double[][] matrix){  for (int i = 0; i < matrix.Length; i++)  {    for (int j = 0; j < matrix[0].Length; j++)    {      if (Math.Abs(matrix[i][j]) <= 0.00001)      {        matrix[i][j] = 0;      }    }  }}/// <summary>/// 計算行最簡矩陣的秩/// </summary>/// <param name="matrix"></param>/// <returns></returns>private static int Operation4(double[][] matrix){  int rank = -1;  bool isAllZero = true;  for (int i = 0; i < matrix.Length; i++)  {    isAllZero = true;    //查看當前行有沒有0    for (int j = 0; j < matrix[0].Length; j++)    {      if (matrix[i][j] != 0)      {        isAllZero = false;        break;      }    }    //若第i行全為0，則矩陣的秩為i    if (isAllZero)    {      rank = i;      break;    }  }  //滿秩矩陣的情況  if (rank == -1)  {    rank = matrix.Length;  }  return rank;}

static void Main(string[] args){  //示例矩陣1：秩為3  double[][] matrix1 = new double[][]   {    new double[] { 1, 1, 1 },    new double[] { 1, 1, 0 },    new double[] { 0, 1, 1 }   };  Console.WriteLine(Rank(matrix1));  //示例矩陣2：秩為3  double[][] matrix2 = new double[][]   {    new double[] { 3, 2, 0, 5, 0 },     new double[] { 3, -2, 3, 6, -1 },    new double[] { 2, 0, 1, 5, -3 },    new double[] { 1, 6, -4, -1, 4 }   };  Console.WriteLine(Rank(matrix2));  //示例矩陣3：秩為3  double[][] matrix3 = new double[][]   {    new double[] { 2, 3, 1, -3, -7 },     new double[] { 1, 2, 0, -2, -4 },    new double[] { 3, -2, 8, 3, 0 },    new double[] { 2, -3, 7, 4, 3 }  };  Console.WriteLine(Rank(matrix3));  Console.ReadLine();}