Find if given matrix is Toeplitz or not

Given a square matrix, find if it’s a Toeplitz matrix or not. A Toeplitz (or diagonal-constant) matrix is a matrix in which each descending diagonal from left to right is constant, i.e., all elements in a diagonal are same.
In general, any n×n matrix mat[][] is a Toeplitz matrix if every cell mat[i][j] is same as mat[i-1][j-1], mat[i+1][j+1], mat[i-2][j-2], mat[i+2][j+2], .. for every cell mat[i][j] and all the valid cells mat[i+k][j+k] or mat[i-k][j-k] 

Examples :

Input: mat[N][N] = {{ 6, 7, 8},
                    { 4, 6, 7},
                    { 1, 4, 6}},
Output : True;
Values in all diagonals are same.

Input: mat[N][N] = {{ 6, 7, 8, 9 },
                    { 4, 6, 7, 8 },
                    { 1, 4, 6, 7 },
                    { 0, 1, 4, 6 }};
Output : True;

Input: mat[N][N] = {{ 6, 3, 8},
                    { 4, 9, 7},
                    { 1, 4, 6}},
Output : False;

The idea is very simple. For each element of first row and first column(or last row and last column) in the matrix, we check if descending diagonal starting from that element have same values or not. If we found any diagonal having different values, we return false.

Below is the implementation of the above code:

C++

// C++ program to check whether given matrix
// is a Toeplitz matrix or not
#include <iostream>
using namespace std;
#define N 5
#define M 4

// Function to check if all elements present in
// descending diagonal starting from position
// (i, j) in the matrix are all same or not
bool checkDiagonal(int mat[N][M], int i, int j)
{
    int res = mat[i][j];
    while (++i < N && ++j < M)
    {
        // mismatch found
        if (mat[i][j] != res)
            return false;
    }

    // we only reach here when all elements
    // in given diagonal are same
    return true;
}

// Function to check whether given matrix is a
// Toeplitz matrix or not
bool isToeplitz(int mat[N][M])
{
    // do for each element in first row
    for (int i = 0; i < M; i++)
    {
        // check descending diagonal starting from
        // position (0, j) in the matrix
        if (!checkDiagonal(mat, 0, i))
            return false;
    }

    // do for each element in first column
    for (int i = 1; i < N; i++)
    {
        // check descending diagonal starting from
        // position (i, 0) in the matrix
        if (!checkDiagonal(mat, i, 0))
            return false;
    }

    // we only reach here when each descending
    // diagonal from left to right is same
    return true;
}

// Driver code
int main()
{
    int mat[N][M] = { { 6, 7, 8, 9 },
                      { 4, 6, 7, 8 },
                      { 1, 4, 6, 7 },
                      { 0, 1, 4, 6 },
                      { 2, 0, 1, 4 } };

    // Function call
    if (isToeplitz(mat))
        cout << "Matrix is a Toeplitz ";
    else
        cout << "Matrix is not a Toeplitz ";

    return 0;
}

Java

// Java program to check whether given matrix
// is a Toeplitz matrix or not
import java.io.*;

class GFG
{
    public static int N = 5;
    public static int M = 4;

    // Function to check if all elements present in
    // descending diagonal starting from position
    // (i, j) in the matrix are all same or not
    static boolean checkDiagonal(int mat[][], int i, int j)
    {
        int res = mat[i][j];
        while (++i < N && ++j < M)
        {
            // mismatch found
            if (mat[i][j] != res)
                return false;
        }

        // we only reach here when all elements
        // in given diagonal are same
        return true;
    }

    // Function to check whether given matrix is a
    // Toeplitz matrix or not
    static boolean isToeplitz(int mat[][])
    {
        // do for each element in first row
        for (int i = 0; i < M; i++)
        {
            // check descending diagonal starting from
            // position (0, j) in the matrix
            if (!checkDiagonal(mat, 0, i))
                return false;
        }

        // do for each element in first column
        for (int i = 1; i < N; i++)
        {
            // check descending diagonal starting from
            // position (i, 0) in the matrix
            if (!checkDiagonal(mat, i, 0))
                return false;
        }

        // we only reach here when each descending
        // diagonal from left to right is same
        return true;
    }

    // Driver code
    public static void main(String[] args)
    {
        int mat[][] = { { 6, 7, 8, 9 },
                        { 4, 6, 7, 8 },
                        { 1, 4, 6, 7 },
                        { 0, 1, 4, 6 },
                        { 2, 0, 1, 4 } };

        // Function call
        if (isToeplitz(mat))
            System.out.println("Matrix is a Toeplitz ");
        else
            System.out.println("Matrix is not a Toeplitz ");
    }
}

// This code is contributed by Pramod Kumar

Python3

# Python3 program to check whether given
# matrix is a Toeplitz matrix or not
N = 5
M = 4

# Function to check if all elements present in
# descending diagonal starting from position
# (i, j) in the matrix are all same or not


def checkDiagonal(mat, i, j):
    res = mat[i][j]
    i += 1
    j += 1

    while (i < N and j < M):

        # mismatch found
        if (mat[i][j] != res):
            return False

        i += 1
        j += 1

    # we only reach here when all elements
    # in given diagonal are same
    return True

# Function to check whether given
# matrix is a Toeplitz matrix or not


def isToeplitz(mat):

    # do for each element in first row
    for j in range(M):

        # check descending diagonal starting from
        # position (0, j) in the matrix
        if not(checkDiagonal(mat, 0, j)):
            return False

    # do for each element in first column
    for i in range(1, N):

        # check descending diagonal starting
        # from position (i, 0) in the matrix
        if not(checkDiagonal(mat, i, 0)):
            return False

    return True


# Driver Code
if __name__ == "__main__":

    mat = [[6, 7, 8, 9],
           [4, 6, 7, 8],
           [1, 4, 6, 7],
           [0, 1, 4, 6],
           [2, 0, 1, 4]]

    # Function call
    if(isToeplitz(mat)):
        print("Matrix is a Toeplitz")
    else:
        print("Matrix is not a Toeplitz")

# This code is contributed by Jasmine K Grewal

C#

// C# program to check whether given matrix
// is a Toeplitz matrix or not
using System;

class GFG {

    public static int N = 5;
    public static int M = 4;

    // Function to check if all elements present in
    // descending diagonal starting from position
    // (i, j) in the matrix are all same or not
    static bool checkDiagonal(int[, ] mat, int i, int j)
    {

        int res = mat[i, j];
        while (++i < N && ++j < M) {

            // mismatch found
            if (mat[i, j] != res)
                return false;
        }

        // we only reach here when all elements
        // in given diagonal are same
        return true;
    }

    // Function to check whether given matrix is a
    // Toeplitz matrix or not
    static bool isToeplitz(int[, ] mat)
    {

        // do for each element in first row
        for (int i = 0; i < M; i++) {

            // check descending diagonal starting from
            // position (0, j) in the matrix
            if (!checkDiagonal(mat, 0, i))
                return false;
        }

        // do for each element in first column
        for (int i = 1; i < N; i++) {

            // check descending diagonal starting from
            // position (i, 0) in the matrix
            if (!checkDiagonal(mat, i, 0))
                return false;
        }

        // we only reach here when each descending
        // diagonal from left to right is same
        return true;
    }

    // Driver code
    public static void Main()
    {
        int[, ] mat = { { 6, 7, 8, 9 },
                        { 4, 6, 7, 8 },
                        { 1, 4, 6, 7 },
                        { 0, 1, 4, 6 },
                        { 2, 0, 1, 4 } };

        // Function call
        if (isToeplitz(mat))
            Console.WriteLine("Matrix is a Toeplitz ");
        else
            Console.WriteLine("Matrix is not a Toeplitz ");
    }
}

// This code is contributed by KRV.

PHP

<?php
// PHP program to check whether
// given matrix is a Toeplitz
// matrix or not

// Function to check if all
// elements present in descending
// diagonal starting from position
// (i, j) in the matrix are all
// same or not
function checkDiagonal($mat, $i, $j)
{
    $N = 5; $M = 4;
    $res = $mat[$i][$j];
    while (++$i < $N && ++$j < $M)
    {
        // mismatch found
        if ($mat[$i][$j] != $res)
            return false;
    }

    // we only reach here when
    // all elements in given
    // diagonal are same
    return true;
}

// Function to check whether
// given matrix is a
// Toeplitz matrix or not
function isToeplitz($mat)
{
    $N = 5; $M = 4;
    // do for each element in first row
    for ($i = 0; $i < $M; $i++)
    {
        // check descending diagonal
        // starting from position
        // (0, j) in the matrix
        if (!checkDiagonal($mat, 0, $i))
            return false;
    }

    // do for each element
    // in first column
    for ($i = 1; $i < $N; $i++)
    {
        // check descending diagonal
        // starting from position
        // (i, 0) in the matrix
        if (!checkDiagonal($mat, $i, 0))
            return false;
    }

    // we only reach here when
    // each descending diagonal
    // from left to right is same
    return true;
}

// Driver code
$mat = array(array( 6, 7, 8, 9 ),
             array( 4, 6, 7, 8 ),
             array( 1, 4, 6, 7 ),
             array( 0, 1, 4, 6 ),
             array( 2, 0, 1, 4 ));

// Function call
if (isToeplitz($mat))
    echo "Matrix is a Toeplitz ";
else
    echo "Matrix is not a Toeplitz ";

// This code is contributed
// by nitin mittal.
?>

Javascript

<script>

    // Javascript program to check whether given matrix
    // is a Toeplitz matrix or not

    let N = 5;
    let M = 4;

    // Function to check if all elements present in
    // descending diagonal starting from position
    // (i, j) in the matrix are all same or not
    function checkDiagonal(mat, i, j)
    {
        let res = mat[i][j];
        while (++i < N && ++j < M)
        {
            // mismatch found
            if (mat[i][j] != res)
                return false;
        }

        // we only reach here when all elements
        // in given diagonal are same
        return true;
    }

    // Function to check whether given matrix is a
    // Toeplitz matrix or not
    function isToeplitz(mat)
    {
        // do for each element in first row
        for (let i = 0; i < M; i++)
        {
            // check descending diagonal starting from
            // position (0, j) in the matrix
            if (!checkDiagonal(mat, 0, i))
                return false;
        }

        // do for each element in first column
        for (let i = 1; i < N; i++)
        {
            // check descending diagonal starting from
            // position (i, 0) in the matrix
            if (!checkDiagonal(mat, i, 0))
                return false;
        }

        // we only reach here when each descending
        // diagonal from left to right is same
        return true;
    }

    let mat = [ [ 6, 7, 8, 9 ],
               [ 4, 6, 7, 8 ],
               [ 1, 4, 6, 7 ],
               [ 0, 1, 4, 6 ],
               [ 2, 0, 1, 4 ] ];

    // Function call
    if (isToeplitz(mat))
        document.write("Matrix is a Toeplitz ");
    else
        document.write("Matrix is not a Toeplitz ");

</script>

Output

Matrix is a Toeplitz 

The time complexity of this solution would be O(mn) where m is number of rows and n is number of columns as we are traversing through each element of the matrix.
Auxiliary Space: O(1), since no extra space has been taken.

This approach can further be improved:

As we can see that we are traversing through entire matrix so we will keep track of each element if it is equal to its diagonally above element or not. If any element fails this condition this means matrix is not “Toeplitz” so return false or if all elements pass this condition then return true.

C++

#include <iostream>
using namespace std;
#define N 5
#define M 4
bool isToeplitz(int matrix[N][M])
{
    for (int i = 1; i < N; i++) {
        for (int j = 1; j < M; j++) {
            // check if each index is equal to its
            // diagonally upper index if not return false
            if (matrix[i][j] != matrix[i - 1][j - 1])
                return false;
        }
    }
    return true;
}
// Driver code
int main()
{
    int mat[N][M] = { { 6, 7, 8, 9 },
                      { 4, 6, 7, 8 },
                      { 1, 4, 6, 7 },
                      { 0, 1, 4, 6 },
                      { 2, 0, 1, 4 } };

    // Function call
    if (isToeplitz(mat))
        cout << "Matrix is a Toeplitz ";
    else
        cout << "Matrix is not a Toeplitz ";

    return 0;
}
//This code is contributed by Harsh Singh NIT Sgr

Java

// Java program to check whether given matrix
// is a Toeplitz matrix or not
import java.io.*;

class GFG {
    public static int N = 5;
    public static int M = 4;

    // Function to check whether given matrix is a
    // Toeplitz matrix or not
    static boolean isToeplitz(int mat[][])
    {
        for (int i = 1; i < N; i++) {
            for (int j = 1; j < M; j++) {
                // check if each index is equal to its
                // diagonally upper index if not return false
                if (mat[i][j] != mat[i - 1][j - 1]) {
                    return false;
                }
            }
        }
        return true;
    }

    // Driver code
    public static void main(String[] args)
    {
        int mat[][] = { { 6, 7, 8, 9 },
                        { 4, 6, 7, 8 },
                        { 1, 4, 6, 7 },
                        { 0, 1, 4, 6 },
                        { 2, 0, 1, 4 } };

        // Function call
        if (isToeplitz(mat))
            System.out.println("Matrix is a Toeplitz ");
        else
            System.out.println("Matrix is not a Toeplitz ");
    }
}

// This code is contributed by Harsh Singh NIT Sgr

Python3

N=5;
M=4;

def isToeplitz( matrix):
    for i in range(1,N):
        for j in range(1,M):
            # check if each index is equal to its
            # diagonally upper index if not return false
            if (matrix[i][j] != matrix[i - 1][j - 1]):
                return False;

    return True;

# Driver code
mat = [ [ 6, 7, 8, 9 ],
                  [ 4, 6, 7, 8 ],
                  [ 1, 4, 6, 7 ],
                  [ 0, 1, 4, 6 ],
                  [ 2, 0, 1, 4 ] ];

# Function call
if (isToeplitz(mat)):
    print("Matrix is a Toeplitz ");
else:
    print("Matrix is not a Toeplitz ");

C#

// C# program to check whether given matrix
// is a Toeplitz matrix or not

using System;

class GFG {
    public static int N = 5;
    public static int M = 4;

    // Function to check whether given matrix is a
    // Toeplitz matrix or not
    static bool isToeplitz(int[, ] mat)
    {
        for (int i = 1; i < N; i++) {
            for (int j = 1; j < M; j++) {
                // check if each index is equal to its
                // diagonally upper index if not return
                // false
                if (mat[i, j] != mat[i - 1, j - 1]) {
                    return false;
                }
            }
        }
        return true;
    }

    // Driver code
    public static void Main(string[] args)
    {
        int[, ] mat = { { 6, 7, 8, 9 },
                        { 4, 6, 7, 8 },
                        { 1, 4, 6, 7 },
                        { 0, 1, 4, 6 },
                        { 2, 0, 1, 4 } };

        // Function call
        if (isToeplitz(mat))
            Console.WriteLine("Matrix is a Toeplitz ");
        else
            Console.WriteLine("Matrix is not a Toeplitz ");
    }
}

// This code is contributed by Karandeep1234

Javascript

const N = 5
const M = 4
function isToeplitz(matrix)
{
    for (let i = 1; i < N; i++) {
        for (let j = 1; j < M; j++) {
            // check if each index is equal to its
            // diagonally upper index if not return false
            if (matrix[i][j] != matrix[i - 1][j - 1])
                return false;
        }
    }
    return true;
}
// Driver code
    let mat = [ [ 6, 7, 8, 9 ],
                      [ 4, 6, 7, 8 ],
                      [ 1, 4, 6, 7 ],
                      [ 0, 1, 4, 6 ],
                      [ 2, 0, 1, 4 ] ];

    // Function call
    if (isToeplitz(mat))
        document.write("Matrix is a Toeplitz ");
    else
        document.write("Matrix is not a Toeplitz ");

Output

Matrix is a Toeplitz 

Time complexity of this solution would be O(mn) where m is number of rows and n is number of columns as we are traversing through each element of the matrix.
Auxiliary Space: O(1), since no extra space has been taken.

Hashing based approach:

Consider an element at index (i, j) of matrix of dimension (m, n). For the matrix to be diagonal-constant, all the elements in the diagonal must be same. Consider the diagonal containing this (i, j) element. The other elements in this diagonal will have their index of the form (i+k, j+k) or (i-k, j-k). Notice that whatever x-value and y-value of these indexes are, their difference is always the same. i.e. (i+k)-(j+k) == (i-k)-(j-k) == i-j. 

The diagram below gives a better visualization of this idea. Consider the diagonal coloured yellow. The difference between x-value and y-value of any index on this diagonal is 2 (2-0, 3-1, 4-2, 5-3). Same can be observed for all body diagonals.

Index of a Toeplitz matrix

For red-coloured diagonal, difference is 3. For green-coloured diagonal, difference is 0. For orange-coloured diagonal, difference is -2 and so on…

The idea is to exploit the fact that for a Toeplitz matrix, these individual index differences for particular diagonals will be unique. And since it is a constant-diagonal matrix, for all these unique keys, there should be unique values same as any element on that diagonal. So, we create a HashMap to store these (key, value) pairs. At any moment if we encounter a value, that is different from it’s corresponding stored key value, we return false. 

Below is the implementation of the above code:

C++

// C++ program to check whether given
// matrix is a Toeplitz matrix or not
#include <bits/stdc++.h>
using namespace std;

bool isToeplitz(vector<vector<int>> matrix)
{

    // row = number of rows
    // col = number of columns
    int row = matrix.size();
    int col = matrix[0].size();

    // HashMap to store key,value pairs
    map<int, int> Map;

    for(int i = 0; i < row; i++)
    {
        for(int j = 0; j < col; j++)
        {
            int key = i - j;

            // If key value exists in the hashmap,
            if (Map[key])
            {

                // We check whether the current
                // value stored in this key
                // matches to element at current
                // index or not. If not, return
                // false
                if (Map[key] != matrix[i][j])
                    return false;
            }

            // Else we put key,value pair in hashmap
            else
            {
                Map[i - j] = matrix[i][j];
            }
        }
    }
    return true;
}

// Driver code
int main()
{
    vector<vector<int>> matrix = { { 12, 23, -32 },
                                   { -20, 12, 23 },
                                   { 56, -20, 12 },
                                   { 38, 56, -20 } };

    // Function call
    string result = (isToeplitz(matrix)) ? "Yes" : "No";

    cout << result;

    return 0;
}

// This code is contributed by divyesh072019

Java

// JAVA program to check whether given matrix
// is a Toeplitz matrix or not

import java.util.*;

class GFG {

    static boolean isToeplitz(int[][] matrix)
    {
        // row = number of rows
        // col = number of columns
        int row = matrix.length;
        int col = matrix[0].length;
        // HashMap to store key,value pairs
        HashMap<Integer, Integer> map
            = new HashMap<Integer, Integer>();
        for (int i = 0; i < row; i++)
        {
            for (int j = 0; j < col; j++)
            {
                int key = i - j;
                // if key value exists in the hashmap,
                if (map.containsKey(key))
                {
                    // we check whether the current value
                    // stored in this key matches to element
                    // at current index or not. If not,
                    // return false
                    if (map.get(key) != matrix[i][j])
                        return false;
                }
                // else we put key,value pair in hashmap
                else {
                    map.put(i - j, matrix[i][j]);
                }
            }
        }
        return true;
    }

    // Driver Code
    public static void main(String[] args)
    {
        int[][] matrix = { { 12, 23, -32 },
                           { -20, 12, 23 },
                           { 56, -20, 12 },
                           { 38, 56, -20 } };

        // Function call
        String result = (isToeplitz(matrix)) ? "Yes" : "No";
        System.out.println(result);
    }
}

Python3

# Python3 program to check whether given matrix
# is a Toeplitz matrix or not

def isToeplitz(matrix):

    # row = number of rows
    # col = number of columns
    row = len(matrix)
    col = len(matrix[0])

    # dictionary to store key,value pairs
    map = {}
    for i in range(row):
        for j in range(col):
            key = i-j

            # if key value exists in the map,
            if (key in map):

                # we check whether the current value stored
                # in this key matches to element at current
                # index or not. If not, return false
                if (map[key] != matrix[i][j]):
                    return False

            # else we put key,value pair in map
            else:
                map[key] = matrix[i][j]
    return True

# Driver Code
if __name__ == "__main__":
    matrix = [[12, 23, -32], [-20, 12, 23], [56, -20, 12], [38, 56, -20]]

    # Function call
    if (isToeplitz(matrix)):
        print("Yes")
    else:
        print("No")

C#

// C# program to check whether given
// matrix is a Toeplitz matrix or not
using System;
using System.Collections.Generic;

class GFG{

static bool isToeplitz(int[,] matrix)
{

    // row = number of rows
    // col = number of columns
    int row = matrix.GetLength(0);
    int col = matrix.GetLength(1);

    // HashMap to store key,value pairs
    Dictionary<int,
               int> map = new Dictionary<int,
                                         int>();

    for(int i = 0; i < row; i++)
    {
        for(int j = 0; j < col; j++)
        {

            int key = i - j;

            // If key value exists in the hashmap,
            if (map.ContainsKey(key))
            {

                // We check whether the current value
                // stored in this key matches to element
                // at current index or not. If not,
                // return false
                if (map[key] != matrix[i, j])
                    return false;
            }

            // Else we put key,value pair in hashmap
            else
            {
                map.Add(i - j, matrix[i, j]);
            }
        }
    }
    return true;
}

// Driver code
static void Main()
{
    int[,] matrix = { { 12, 23, -32 },
                      { -20, 12, 23 },
                      { 56, -20, 12 },
                      { 38, 56, -20 } };

    // Function call
    string result = (isToeplitz(matrix)) ?
                    "Yes" : "No";

    Console.WriteLine(result);
}
}

// This code is contributed by divyeshrabadiya07

Javascript

<script>

    // JavaScript program to check whether given
    // matrix is a Toeplitz matrix or not
    function isToeplitz(matrix)
    {

        // row = number of rows
        // col = number of columns
        let row = matrix.length;
        let col = matrix[0].length;

        // HashMap to store key,value pairs
        let map = new Map();

        for(let i = 0; i < row; i++)
        {
            for(let j = 0; j < col; j++)
            {
                let key = i - j;

                // If key value exists in the hashmap,
                if (map.has(key))
                {

                    // We check whether the current
                    // value stored in this key
                    // matches to element at current
                    // index or not. If not, return
                    // false
                    if (map.get(key) != matrix[i][j])
                        return false;
                }

                // Else we put key,value pair in hashmap
                else
                {
                    map.set(i - j, matrix[i][j]);
                }
            }
        }
        return true;
    }

    // Driver code

    let matrix = [ [ 12, 23, -32 ],
                  [ -20, 12, 23 ],
                  [ 56, -20, 12 ],
                  [38, 56, -20 ] ];

    // Function call
    let result = (isToeplitz(matrix)) ? "Yes" : "No";

    document.write(result);

</script>

Output

Yes

Time Complexity: O(mn), where m is number of rows and n is number of columns.
Auxiliary Space: O(m+n), because at worst case, if a matrix is Toeplitz, we have store exactly (m+n-1) key, value pairs. (In first row we have n distinct keys and then for next each m-1 rows, we keep adding one unique key to the map.

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