# 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:

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 ``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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