Hilbert Matrix

A Hilbert Matrix is a square matrix whose each element is a unit fraction.
Properties:

1. It is a symmetric matrix.
2. Its determinant value is always positive.

   Examples:

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   Input : N = 2
   Output : 1    0.5
            0.5  0.33                   
   
   Input : N = 3
   Output : 1.0000    0.5000    0.3333
            0.5000    0.3333    0.2500
            0.3333    0.2500    0.2000
   

   Recommended: Please try your approach on {IDE} first, before moving on to the solution.

   Mathematically, Hilbert Matrix can be formed by the given formula:

   
   
   
   

    
   Let H be a Hilbert Matrix of NxN.
   Then
   H(i, j) = 1/(i+j-1)
   

   Below is the basic implementation of the above formula.
   

   C++

   
   

   
   
   

   // C++ program for Hilbert Matrix #include <bits/stdc++.h> using namespace std;    // Function that generates a Hilbert matrix void printMatrix(int n) {     float H[n][n];        for (int i = 0; i < n; i++) {         for (int j = 0; j < n; j++) {                // using the formula to generate             // hilbert matrix             H[i][j] = (float)1.0 /                       ((i + 1) + (j + 1) - 1.0);         }     }        for (int i = 0; i < n; i++) {         for (int j = 0; j < n; j++)              cout << H[i][j] << " ";                 cout << endl;     } }    // driver function int main() {     int n = 3;     printMatrix(n);     return 0; }

   Java

   
   

   
   
   

   // Java program for // Hilbert Matrix import java.io.*;    class GFG  {        // Function that generates  // a Hilbert matrix static void printMatrix(int n) {     float H[][] = new float[n][n];        for (int i = 0; i < n; i++)     {         for (int j = 0; j < n; j++)          {                // using the formula              // to generate             // hilbert matrix             H[i][j] = (float)1.0 /                        ((i + 1) + (j + 1) -                        (float)1.0);         }     }        for (int i = 0; i < n; i++)      {         for (int j = 0; j < n; j++)              System.out.print(H[i][j] + " ");          System.out.println();     } }    // Driver code public static void main (String[] args)  {     int n = 3;     printMatrix(n); } }    // This code is contributed  // by anuj_67.

   C#

   
   

   
   
   

   // C# program for Hilbert Matrix using System;    class GFG  {        // Function that generates  // a Hilbert matrix static void printMatrix(int n) {     float[,] H = new float[n, n];        for (int i = 0; i < n; i++)     {         for (int j = 0; j < n; j++)          {                // using the formula to generate             // hilbert matrix             H[i, j] = (float)1.0 /                       ((i + 1) + (j + 1) -                        (float)1.0);         }     }        for (int i = 0; i < n; i++)      {         for (int j = 0; j < n; j++)              Console.Write(H[i, j] + " ");          Console.WriteLine("");     } }    // Driver code public static void Main()  {     int n = 3;     printMatrix(n); } }    // This code is contributed  // by mits

   Output:

   1 0.5 0.333333 
   0.5 0.333333 0.25 
   0.333333 0.25 0.2
   

   
   
   
   
   
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