# 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.
3. 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 ` `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
```

My Personal Notes arrow_drop_up Maths is the language of nature

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