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] == a_{i + j}. Now, a_{i + j} can be define as:

         m[i+j][0], if i + j < n
a_{i + j} = 
         m[i + j - n + 1][n-1], otherwise

C++

// C++ Program to check if given matrix is 
// Hankel Matrix or not. 
#include <bits/stdc++.h> 
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

<?php 
// PHP Program to check if given matrix is 
// Hankel Matrix or not. 
$N = 4; 
  
// Function to check if  
// given matrix is Hankel 
// Matrix or not. 
function checkHankelMatrix( $n, $m) 
{ 
      
    // for each row 
    for($i = 0; $i < $n; $i++) { 
  
        // for each column 
        for ($j = 0;$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; 
} 
  
    // Driver code 
    $n = 4; 
    $m = array(array( 1, 2, 3, 5 ), 
               array( 2, 3, 5, 8 ), 
               array( 3, 5, 8, 0 ), 
               array( 5, 8, 0, 9 )); 
    if(checkHankelMatrix($n, $m)) 
        echo "Yes"; 
    else
        echo "No"; 
  
// This code is contributed by Anuj_67. 
?> 

Output:

Yes

Complexity : O(n + n)

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C++

Java

Python 3

C#

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