Hilbert Matrix

Difficulty Level : Easy
Last Updated : 09 Nov, 2020

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

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

Examples:

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

C++

Java

C#