C#计算矩阵的秩实例分析

/// <summary>
/// 计算矩阵的秩
/// </summary>
/// <param name="matrix">矩阵</param>
/// <returns></returns>
private static int Rank(double[][] matrix)
{
  //matrix为空则直接默认已经是最简形式
  if (matrix == null || matrix.Length == 0) return 0;
  //复制一个matrix到copy，之后因计算需要改动矩阵时并不改动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];
    }
  }
  //先以最左侧非零项的位置进行行排序
  Operation1(copy);
  //循环化简矩阵
  while (!isFinished(copy))
  {
    Operation2(copy);
    Operation1(copy);
  }
  //过于趋近0的项，视作0，减小误差
  Operation3(copy);
  //行最简矩阵的秩即为所求
  return Operation4(matrix);
}
/// <summary>
/// 判断矩阵是否变换到最简形式（非零行数达到最少）
/// </summary>
/// <param name="matrix"></param>
/// <returns>true:</returns>
private static bool isFinished(double[][] matrix)
{
  //统计每行第一个非零元素的出现位置
  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现位置必须在前面行的后面，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（按左侧最前非零位位置自上而下升序排列）
/// </summary>
/// <param name="matrix">矩阵</param>
private static void Operation1(double[][] matrix)
{
  //统计每行第一个非零元素的出现位置
  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现位置升序排列
  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>
/// 行初等变换（左侧最前非零位位置最靠前的行，只保留一个）
/// </summary>
/// <param name="matrix">矩阵</param>
private static void Operation2(double[][] matrix)
{
  //统计每行第一个非零元素的出现位置
  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>
/// 将和0非常接近的数字视为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>
/// 计算行最简矩阵的秩
/// </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;
    //查看当前行有没有0
    for (int j = 0; j < matrix[0].Length; j++)
    {
      if (matrix[i][j] != 0)
      {
        isAllZero = false;
        break;
      }
    }
    //若第i行全为0，则矩阵的秩为i
    if (isAllZero)
    {
      rank = i;
      break;
    }
  }
  //满秩矩阵的情况
  if (rank == -1)
  {
    rank = matrix.Length;
  }
  return rank;
}

static void Main(string[] args)
{
  //示例矩阵1：秩为3
  double[][] matrix1 = new double[][] 
  {
    new double[] { 1, 1, 1 },
    new double[] { 1, 1, 0 },
    new double[] { 0, 1, 1 } 
  };
  Console.WriteLine(Rank(matrix1));
  //示例矩阵2：秩为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));
  //示例矩阵3：秩为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();
}