# 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

Below is the implementation of the above approach:

 // 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 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 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# 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.



Output
Yes

Time Complexity: O(N2)
Auxiliary Space: O(1)

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