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:

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

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.

Python3

# Python3 program for Hilbert Matrix

# Function that generates a Hilbert matrix
def printMatrix(n):
    H = [[None for _ in range(n)] for __ in range(n)]

    for i in range(n):
        for j in range(n):
            # using the formula to generate
            # hilbert matrix
            H[i][j] = 1 / ((i + 1) + (j + 1) - 1)

    for i in range(n):
        print(*H[i])

# driver function
n = 3
printMatrix(n)

# This code is contributed by phasing17

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

Javascript

// JavaScript program for Hilbert Matrix


// Function that generates a Hilbert matrix
function printMatrix(n)
{
    let H = new Array(n);

    for (var i = 0; i < n; i++) {
        H[i] = new Array(n);
        for (var j = 0; j < n; j++) {

            // using the formula to generate
            // hilbert matrix
            H[i][j] = 1.0 /
                    ((i + 1) + (j + 1) - 1.0);
        }
    }

    for (var i = 0; i < n; i++) {
        for (var j = 0; j < n; j++)
            process.stdout.write(H[i][j] + " ");
        process.stdout.write("\n");
    }
}

// driver function
n = 3;
printMatrix(n);



// This code is contributed by phasing17

Output

1 0.5 0.333333 
0.5 0.333333 0.25 
0.333333 0.25 0.2 

Time complexity: O(n²)
Auxiliary space: O(n²)