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C#，数值计算，矩阵的行列式（Determinant）、伴随矩阵（Adjoint）与逆矩阵（Inverse）的算法与源代码

news 来源：原创 2024/5/20 19:16:29

本文发布矩阵（Matrix）的一些初级算法。

一、矩阵的行列式（Determinant）

矩阵行列式是指矩阵的全部元素构成的行列式，设A=(a)是数域P上的一个n阶矩阵，则所有A=(a)中的元素组成的行列式称为矩阵A的行列式，记为|A|或det(A)。若A，B是数域P上的两个n阶矩阵，k是P中的任一个数，则|AB|=|A||B|，|kA|=kⁿ|A|，|A*|=|A|，其中A*是A的伴随矩阵；若A是可逆矩阵，则|A|=|A|。

/// <summary>
/// 计算 A[p,q] 位于 [,]temp 的块辅因子
/// </summary>
/// <param name="matrix"></param>
/// <param name="temp"></param>
/// <param name="p"></param>
/// <param name="q"></param>
/// <param name="n"></param>
private static void BlockCofactor(double[,] matrix, ref double[,] temp, int p, int q, int n)
{
int i = 0;
int j = 0;

   for (int row = 0; row < n; row++)
   {
       for (int col = 0; col < n; col++)
       {
           if (row != p && col != q)
           {
               temp[i, j++] = matrix[row, col];
               if (j == (n - 1))
               {
                   j = 0;
                   i++;
               }
           }
       }
   }
}

/// <summary>
/// 求矩阵行列式（递归算法）
/// </summary>
/// <param name="N"></param>
/// <param name="matrix"></param>
/// <param name="n"></param>
/// <returns></returns>
public static double Determinant(int N, double[,] matrix, int n)
{
   if (n == 1)
   {
       return matrix[0, 0];
   }

   double D = 0.0;
   double[,] temp = new double[N, N];
   int sign = 1;
   for (int f = 0; f < n; f++)
   {
       BlockCofactor(matrix, ref temp, 0, f, n);
       D += sign * matrix[0, f] * Determinant(N, temp, n - 1);
       sign = -sign;
   }
   return D;
}

/// <summary>
/// 计算 A[p,q] 位于 [,]temp 的块辅因子
/// </summary>
/// <param name="matrix"></param>
/// <param name="temp"></param>
/// <param name="p"></param>
/// <param name="q"></param>
/// <param name="n"></param>
private static void BlockCofactor(double[,] matrix, ref double[,] temp, int p, int q, int n)
{int i = 0;int j = 0;for (int row = 0; row < n; row++){for (int col = 0; col < n; col++){if (row != p && col != q){temp[i, j++] = matrix[row, col];if (j == (n - 1)){j = 0;i++;}}}}
}/// <summary>
/// 求矩阵行列式（递归算法）
/// </summary>
/// <param name="N"></param>
/// <param name="matrix"></param>
/// <param name="n"></param>
/// <returns></returns>
public static double Determinant(int N, double[,] matrix, int n)
{if (n == 1){return matrix[0, 0];}double D = 0.0;double[,] temp = new double[N, N];int sign = 1;for (int f = 0; f < n; f++){BlockCofactor(matrix, ref temp, 0, f, n);D += sign * matrix[0, f] * Determinant(N, temp, n - 1);sign = -sign;}return D;
}

二、矩阵的伴随矩阵（Adjoint Matrix）

一个方形矩阵的伴随矩阵是一个类似于逆矩阵的概念。如果二维矩阵可逆，那么它的逆矩阵和它的伴随矩阵之间只差一个系数，对多维矩阵也存在这个规律。然而，伴随矩阵对不可逆的矩阵也有定义，并且不需要用到除法。

/// <summary>
/// 伴随矩阵
/// </summary>
/// <param name="A"></param>
/// <param name="adj"></param>
public static void Adjoint(double[,] matrix, out double[,] adjoint)
{
int N = matrix.GetLength(0);
adjoint = new double[N, N];

   if (N == 1)
   {
       adjoint[0, 0] = 1.0;
       return;
   }

   int sign = 1;
   double[,] temp = new double[N, N];
   for (int i = 0; i < N; i++)
   {
       for (int j = 0; j < N; j++)
       {
           BlockCofactor(matrix, ref temp, i, j, N);
           sign = ((i + j) % 2 == 0) ? 1 : -1;
           adjoint[j, i] = (sign) * (Determinant(N, temp, N - 1));
       }
   }
}

/// <summary>
/// 伴随矩阵
/// </summary>
/// <param name="A"></param>
/// <param name="adj"></param>
public static void Adjoint(double[,] matrix, out double[,] adjoint)
{int N = matrix.GetLength(0);adjoint = new double[N, N];if (N == 1){adjoint[0, 0] = 1.0;return;}int sign = 1;double[,] temp = new double[N, N];for (int i = 0; i < N; i++){for (int j = 0; j < N; j++){BlockCofactor(matrix, ref temp, i, j, N);sign = ((i + j) % 2 == 0) ? 1 : -1;adjoint[j, i] = (sign) * (Determinant(N, temp, N - 1));}}
}

三、矩阵的逆矩阵（Inverse Matrix）

设A是一个n阶矩阵，若存在另一个n阶矩阵B，使得： AB=BA=E ，则称方阵A可逆，并称方阵B是A的逆矩阵。矩阵求逆，即求矩阵的逆矩阵。矩阵是线性代数的主要内容，很多实际问题用矩阵的思想去解既简单又快捷。逆矩阵又是矩阵理论的很重要的内容，逆矩阵的求法自然也就成为线性代数研究的主要内容之一。

/// <summary>
/// 矩阵求逆
/// </summary>
/// <param name="A"></param>
/// <param name="inverse"></param>
/// <returns></returns>
public static bool Inverse(double[,] matrix, out double[,] inverse)
{
int N = matrix.GetLength(0);
inverse = new double[N, N];

   double det = Determinant(N, matrix, N);
   if (det == 0)
   {
       return false;
   }

Adjoint(matrix, out double[,] adj);

   for (int i = 0; i < N; i++)
   {
       for (int j = 0; j < N; j++)
       {
           inverse[i, j] = adj[i, j] / (double)det;
       }
   }
   return true;
}

/// <summary>
/// 矩阵求逆
/// </summary>
/// <param name="A"></param>
/// <param name="inverse"></param>
/// <returns></returns>
public static bool Inverse(double[,] matrix, out double[,] inverse)
{int N = matrix.GetLength(0);inverse = new double[N, N];double det = Determinant(N, matrix, N);if (det == 0){return false;}Adjoint(matrix, out double[,] adj);for (int i = 0; i < N; i++){for (int j = 0; j < N; j++){inverse[i, j] = adj[i, j] / (double)det;}}return true;
}

演算代码：

private void button1_Click(object sender, EventArgs e)
{
   double[,] A = {
       {5, -2, 2, 7},
       {1, 0, 0, 3},
       {-3, 1, 5, 0},
       {3, -1, -9, 4}
   };

double d = Algorithm_Gallery.Determinant(4, A, 4);

   StringBuilder sb = new StringBuilder();
   sb.Append(Welcome());
   sb.AppendLine("1、<b>原始矩阵</b>（Source Matrix）：<br>");
   sb.Append(Algorithm_Gallery.ToHtml(A));
   sb.AppendLine("行列式（Determinant）=" + d + "<br>");

   Algorithm_Gallery.Adjoint(A, out double[,] adj);
   sb.AppendLine("<br>2、<b>伴随矩阵</b>（Adjoint Matrix）：<br>");
   sb.Append(Algorithm_Gallery.ToHtml(adj));

   Algorithm_Gallery.Inverse(A, out double[,] inv);
   sb.AppendLine("<br>3、<b>逆矩阵</b>（Inverse Matrix）：<br>");
   sb.Append(Algorithm_Gallery.ToHtml(inv));
   sb.Append(Bye());
   webBrowser1.DocumentText = sb.ToString();
}

private void button1_Click(object sender, EventArgs e)
{double[,] A = { {5, -2, 2, 7},{1, 0, 0, 3},{-3, 1, 5, 0},{3, -1, -9, 4}};double d = Algorithm_Gallery.Determinant(4, A, 4);StringBuilder sb = new StringBuilder();sb.Append(Welcome());sb.AppendLine("1、<b>原始矩阵</b>（Source Matrix）：<br>");sb.Append(Algorithm_Gallery.ToHtml(A));sb.AppendLine("行列式（Determinant）=" + d + "<br>");Algorithm_Gallery.Adjoint(A, out double[,] adj);sb.AppendLine("<br>2、<b>伴随矩阵</b>（Adjoint Matrix）：<br>");sb.Append(Algorithm_Gallery.ToHtml(adj));Algorithm_Gallery.Inverse(A, out double[,] inv);sb.AppendLine("<br>3、<b>逆矩阵</b>（Inverse Matrix）：<br>");sb.Append(Algorithm_Gallery.ToHtml(inv));sb.Append(Bye());webBrowser1.DocumentText = sb.ToString();
}

打印矩阵的代码：

public static string ToHtml(double[,] y)
{
   int m = y.GetLength(0);
   int n = y.GetLength(1);
   StringBuilder sb = new StringBuilder();
   sb.AppendLine("<style>");
   sb.AppendLine("td { padding:5px;text-align:right; }");
   sb.AppendLine("</style>");
   sb.AppendLine("<table width='100%' border=1 bordercolor='#999999' style='border-collapse:collapse;'>");
   for (int i = 0; i < m; i++)
   {
       sb.AppendLine("<tr>");
       for (int j = 0; j < n; j++)
       {
           sb.AppendLine("<td>" + String.Format("{0:F8}", y[i, j]) + "</td>");
       }
       sb.AppendLine("</tr>");
   }
   sb.AppendLine("</table>");
   return sb.ToString();
}

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public static string ToHtml(double[,] y)
{int m = y.GetLength(0);int n = y.GetLength(1);StringBuilder sb = new StringBuilder();sb.AppendLine("<style>");sb.AppendLine("td { padding:5px;text-align:right; }");sb.AppendLine("</style>");sb.AppendLine("<table width='100%' border=1 bordercolor='#999999' style='border-collapse:collapse;'>");for (int i = 0; i < m; i++){sb.AppendLine("<tr>");for (int j = 0; j < n; j++){sb.AppendLine("<td>" + String.Format("{0:F8}", y[i, j]) + "</td>");}sb.AppendLine("</tr>");}sb.AppendLine("</table>");return sb.ToString();
}