# Check if a given matrix is Hankel or not

Given a matrix m[][] of size n x n. The task is to check whether given matrix is Hankel Matrix or not.

In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a square matrix in which each ascending skew-diagonal from left to right is constant.

Examples:

Input : n = 4,
m[][] = {
{1, 2, 3, 5},
{2, 3, 5, 8},
{3, 5, 8, 0},
{5, 8, 0, 9}
};
Output : Yes
All diagonal {1}, {2, 2}, {3, 3, 3}, {5, 5, 5, 5}, {8, 8, 8}, {9} have constant value.
So given matrix is Hankel Matrix.

Input : n = 3,
m[][] = {
{1, 2, 3},
{2, 3, 5},
{3, 9, 8}
};
Output : No

Observe, for a matrix to be Hankel Matrix, it must be of the form,

```a0  a1  a2  a3
a1  a2  a3  a4
a2  a3  a4  a5
a3  a4  a5  a6
```

Therefore, to check if the given matrix is Hankel Matrix, we need check if each m[i][j] == ai + j. Now, ai + j can be define as:

```         m[i+j][0], if i + j < n
ai + j =
m[i + j - n + 1][n-1], otherwise
```

## C++

 `// C++ Program to check if given matrix is ` `// Hankel Matrix or not. ` `#include ` `using` `namespace` `std; ` `#define N 4 ` ` `  `// Function to check if given matrix is Hankel ` `// Matrix or not. ` `bool` `checkHankelMatrix(``int` `n, ``int` `m[N][N]) ` `{ ` `    ``// for each row ` `    ``for` `(``int` `i = 0; i < n; i++) { ` ` `  `        ``// for each column ` `        ``for` `(``int` `j = 0; j < n; j++) { ` ` `  `            ``// checking if i + j is less than n ` `            ``if` `(i + j < n) { ` ` `  `                ``// checking if the element is equal to the ` `                ``// corresponding diagonal constant ` `                ``if` `(m[i][j] != m[i + j][0]) ` `                    ``return` `false``; ` `            ``} ` `            ``else` `{ ` ` `  `                ``// checking if the element is equal to the ` `                ``// corresponding diagonal constant ` `                ``if` `(m[i][j] != m[i + j - n + 1][n - 1]) ` `                    ``return` `false``; ` `            ``} ` `        ``} ` `    ``} ` ` `  `    ``return` `true``; ` `} ` ` `  `// Drivers code ` `int` `main() ` `{ ` `    ``int` `n = 4; ` `    ``int` `m[N][N] = { ` `        ``{ 1, 2, 3, 5 }, ` `        ``{ 2, 3, 5, 8 }, ` `        ``{ 3, 5, 8, 0 }, ` `        ``{ 5, 8, 0, 9 } ` `    ``}; ` ` `  `    ``checkHankelMatrix(n, m) ? (cout << ``"Yes"``)  ` `                            ``: (cout << ``"No"``); ` `    ``return` `0; ` `} `

## Java

 `// Java Program to check if given matrix is ` `// Hankel Matrix or not. ` `import` `java.io.*; ` `import` `java.util.*; ` ` `  `class` `GFG { ` ` `  `    ``// Function to check if given matrix  ` `    ``// is Hankel Matrix or not. ` `    ``static` `boolean` `checkHankelMatrix(``int` `n, ` `                                 ``int` `m[][]) ` `    ``{ ` `        ``// for each row ` `        ``for` `(``int` `i = ``0``; i < n; i++) { ` `     `  `            ``// for each column ` `            ``for` `(``int` `j = ``0``; j < n; j++) { ` `     `  `                ``// checking if i + j is less ` `                ``// than n ` `                ``if` `(i + j < n) { ` `     `  `                    ``// checking if the element ` `                    ``// is equal to the ` `                    ``// corresponding diagonal ` `                    ``// constant ` `                    ``if` `(m[i][j] != m[i + j][``0``]) ` `                        ``return` `false``; ` `                ``} ` `                ``else` `{ ` `     `  `                    ``// checking if the element ` `                    ``// is equal to the ` `                    ``// corresponding diagonal ` `                    ``// constant ` `                    ``if` `(m[i][j] !=  ` `                       ``m[i + j - n + ``1``][n - ``1``]) ` `                        ``return` `false``; ` `                ``} ` `            ``} ` `        ``} ` `     `  `        ``return` `true``; ` `    ``} ` `     `  `    ``// Drivers code ` `    ``public` `static` `void` `main(String args[]) ` `    ``{ ` `        ``int` `n = ``4``; ` `        ``int` `m[][] = { ` `            ``{ ``1``, ``2``, ``3``, ``5` `}, ` `            ``{ ``2``, ``3``, ``5``, ``8` `}, ` `            ``{ ``3``, ``5``, ``8``, ``0` `}, ` `            ``{ ``5``, ``8``, ``0``, ``9` `} ` `        ``}; ` `     `  `        ``if``(checkHankelMatrix(n, m))  ` `            ``System.out.println(``"Yes"``); ` `        ``else` `            ``System.out.println(``"No"``); ` `    ``} ` `} ` ` `  `// This code is contributed by Anuj_67. `

## Python 3

 `# Python 3 Program to check if given matrix is ` `# Hankel Matrix or not. ` ` `  `N ``=` `4` ` `  `# Function to check if given matrix is Hankel ` `# Matrix or not. ` `def` `checkHankelMatrix(n, m): ` ` `  `    ``# for each row ` `    ``for` `i ``in` `range``( ``0``, n): ` ` `  `        ``# for each column ` `        ``for` `j ``in` `range``( ``0``, n):  ` ` `  `            ``# checking if i + j is less ` `            ``# than n ` `            ``if` `(i ``+` `j < n): ` ` `  `                ``# checking if the element is ` `                ``# equal to the corresponding ` `                ``# diagonal constant ` `                ``if` `(m[i][j] !``=` `m[i ``+` `j][``0``]): ` `                    ``return` `False` `             `  `            ``else` `: ` ` `  `                ``# checking if the element is ` `                ``# equal to the corresponding ` `                ``# diagonal constant ` `                ``if` `(m[i][j] !``=`  `                    ``m[i ``+` `j ``-` `n ``+` `1``][n ``-` `1``]): ` `                    ``return` `False` `             `  `    ``return` `True` ` `  `# Drivers code ` `n ``=` `4` `m ``=``[[``1``, ``2``, ``3``, ``5``,], ` `    ``[``2``, ``3``, ``5``, ``8``,], ` `    ``[``3``, ``5``, ``8``, ``0``,], ` `    ``[``5``, ``8``, ``0``, ``9``]]  ` `(``print``(``"Yes"``) ``if` `checkHankelMatrix(n, m) ` `                      ``else` `print``(``"No"``)) ` ` `  `# This code is contributed by Smitha. `

## C#

 `// C# Program to check if given matrix is ` `// Hankel Matrix or not. ` `using` `System; ` ` `  `class` `GFG { ` ` `  `    ``// Function to check if given matrix  ` `    ``// is Hankel Matrix or not. ` `    ``static` `bool` `checkHankelMatrix(``int` `n, ` `                                ``int` `[,]m) ` `    ``{ ` `        ``// for each row ` `        ``for` `(``int` `i = 0; i < n; i++) { ` `     `  `            ``// for each column ` `            ``for` `(``int` `j = 0; j < n; j++) { ` `     `  `                ``// checking if i + j is less ` `                ``// than n ` `                ``if` `(i + j < n) { ` `     `  `                    ``// checking if the element ` `                    ``// is equal to the ` `                    ``// corresponding diagonal ` `                    ``// constant ` `                    ``if` `(m[i, j] != m[i + j, 0]) ` `                        ``return` `false``; ` `                ``} ` `                ``else` `{ ` `     `  `                    ``// checking if the element ` `                    ``// is equal to the ` `                    ``// corresponding diagonal ` `                    ``// constant ` `                    ``if` `(m[i,j] != m[i + j - n  ` `                                  ``+ 1, n - 1]) ` `                        ``return` `false``; ` `                ``} ` `            ``} ` `        ``} ` `     `  `        ``return` `true``; ` `    ``} ` `     `  `    ``// Drivers code ` `    ``public` `static` `void` `Main() ` `    ``{ ` `        ``int` `n = 4; ` `        ``int` `[,]m = { ` `            ``{ 1, 2, 3, 5 }, ` `            ``{ 2, 3, 5, 8 }, ` `            ``{ 3, 5, 8, 0 }, ` `            ``{ 5, 8, 0, 9 } ` `        ``}; ` `     `  `        ``if``(checkHankelMatrix(n, m))  ` `            ``Console.Write(``"Yes"``); ` `        ``else` `            ``Console.Write(``"No"``); ` `    ``} ` `} ` ` `  `// This code is contributed by Anuj_67. `

## PHP

 ` `

Output:

```Yes
```

Complexity : O(n + n)

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