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Find if given matrix is Toeplitz or not

The problem involves determining whether a given matrix is a Toeplitz matrix or not. A Toeplitz matrix is a special type of matrix in which each diagonal from top-left to bottom-right has the same elements. In other words, the matrix has a constant value along each of its diagonals.

Example

Consider the following matrix:

 1  2  3  4  5
    6  1  2  3  4
    8  6  1  2  3
    7  8  6  1  2
    9  7  8  6  1

In this example, each diagonal from top-left to bottom-right contains the same elements, thus making it a Toeplitz matrix.

Idea to Solve the Problem

The idea to solve this problem is to iterate through the matrix and for each element, check if the diagonal elements starting from that element are the same. If the diagonal elements are the same for all diagonals starting from each element in the first row and the first column, then the matrix is a Toeplitz matrix.

Algorithm

  1. Define a function diagonal that takes the matrix, start row, and start column as parameters.

  2. Initialize a variable element with the value of the matrix element at (start_row, start_col).

  3. Loop through the diagonal elements:

    • For each (i, j) element where i = start_row + 1 and j = start_col + 1, compare the value of matrix[i][j] with element.
    • If they are not equal, return 0, indicating that the diagonal is not the same.

  4. Return 1, indicating that the diagonal is the same.

  5. Define a function is_toeplitz that takes the matrix as a parameter.

  6. Initialize a variable result to 1.

  7. Loop through each column of the first row and check the diagonal using the diagonal function.

  8. Loop through each row of the first column and check the diagonal using the diagonal function.

  9. If result is 1, output "Toeplitz matrix," otherwise, output "Not a Toeplitz matrix."

Pseudocode

diagonal(matrix, start_row, start_col):
    element = matrix[start_row][start_col]
    for i from start_row + 1 to ROW and j from start_col + 1 to COL:
        if matrix[i][j] is not equal to element:
            return 0
    return 1

is_toeplitz(matrix):
    result = 1
    for i from 0 to COL:
        result = diagonal(matrix, 0, i)
    for i from 1 to ROW:
        result = diagonal(matrix, i, 0)
    if result is 1:
        output "Toeplitz matrix"
    else:
        output "Not a Toeplitz matrix"

main:
    matrix = ...  // Define the matrix
    is_toeplitz(matrix)

Code Solution

/*
  C Program 
+ Find if given matrix is Toeplitz or not
*/
#include<stdio.h>
#define ROW 5
#define COL 5

//Function which is check diagonal element values are similar or not
int diagonal(int matrix[ROW][COL],int start_row,int start_col)
{
  
  int status  = 1;
  int element = matrix[start_row][start_col];
  for (int i = start_row+1,j=start_col+1; i < ROW && j<COL; ++i,++j)
  {
    if(matrix[i][j]!=element)
    {
      return 0;
    }
  }
 return 1;
}
//Determine whether matrix is Toeplitz matrix or not
void is_toeplitz(int matrix[ROW][COL])
{
  
  int result=1;

  for (int i = 0; i < COL && result==1; ++i)
  {
    //check upper triangular
    result = diagonal(matrix,0,i);
  }
  for (int i = 1; i < ROW && result==1; ++i)
  {
    //check lower triangular
    result = diagonal(matrix,i,0);
  }

  if(result==1)
  {
    printf("Toeplitz matrix\n");
  }
  else
  {
    printf("Not a Toeplitz matrix\n");
  }
 
}
int main(){

  int arr[ROW][COL]={

    {1,  2,  3,  4,  5},
    {6,  1,  2,  3,  4}, 
    {8,  6,  1,  2,  3},
    {7,  8,  6,  1,  2},
    {9,  7,  8,  6,  1}
  };

  is_toeplitz(arr);

  return 0;
}

Output

Toeplitz matrix
/*
  C++ Program
  Find if given matrix is Toeplitz or not
*/
#include<iostream>
#define ROW 5
#define COL 5
using namespace std;


class MyMatrix {
	public:
		int rows;
	int cols;
	MyMatrix() {
		//Get matrix size
		this->rows = ROW;
		this->cols = COL;
	}
	//Function which is check diagonal element values are similar or not
	bool diagonal(int matrix[][COL], int start_row, int start_col) {
		int status = 1;
		int element = matrix[start_row][start_col];
		for (int i = start_row + 1, j = start_col + 1; i < this->rows && j < this->cols; ++i, ++j) {
			if (matrix[i][j] != element) {
				return false;
			}
		}
		return true;
	}
	// Determine whether matrix is Toeplitz matrix or not
	void is_toeplitz(int matrix[][COL]) {
		bool result = true;
		for (int i = 0; i < this->cols && result == true; ++i) {
			//check upper triangular
			result = this->diagonal(matrix, 0, i);
		}
		for (int i = 1; i < this->rows && result == true; ++i) {
			//check lower triangular
			result = this->diagonal(matrix, i, 0);
		}
		if (result == true) {
			cout << "Toeplitz matrix\n";
		} else {
			cout << "Not a Toeplitz matrix\n";
		}
	}
};
int main() {
	int matrix[][COL] = {
		{
			1,
			2,
			3,
			4,
			5
		},
		{
			6,
			1,
			2,
			3,
			4
		},
		{
			8,
			6,
			1,
			2,
			3
		},
		{
			7,
			8,
			6,
			1,
			2
		},
		{
			9,
			7,
			8,
			6,
			1
		}
	};
	MyMatrix obj ;
	obj.is_toeplitz(matrix);
	return 0;
}

Output

Toeplitz matrix
using System;

/*
  C# Program
  Find if given matrix is Toeplitz or not
*/

public class MyMatrix {
	int rows;
	int cols;
	MyMatrix(int[,] matrix) {
		//Get matrix size
		this.rows = matrix.GetLength(0);
		this.cols = matrix.GetLength(1);
	}
	//Function which is check diagonal element values are similar or not
	public Boolean diagonal(int[,] matrix, int start_row, int start_col) {
	
		int element = matrix[start_row,start_col];
		for (int i = start_row + 1, j = start_col + 1; i < this.rows && j < this.cols; ++i, ++j) {
			if (matrix[i,j] != element) {
				return false;
			}
		}
		return true;
	}
	// Determine whether matrix is Toeplitz matrix or not
	public void is_toeplitz(int[,] matrix) {
		Boolean result = true;
		for (int i = 0; i < this.cols && result == true; ++i) {
			//check upper triangular
			result = diagonal(matrix, 0, i);
		}
		for (int i = 1; i < this.rows && result == true; ++i) {
			//check lower triangular
			result = diagonal(matrix, i, 0);
		}
		if (result == true) {
			Console.Write("Toeplitz matrix\n");
		} else {
			Console.Write("Not a Toeplitz matrix\n");
		}
	}
	public static void Main(String[] args) {
		int[,]
		//Define matrix element
		matrix = {
			{
				1,
				2,
				3,
				4,
				5
			},
			{
				6,
				1,
				2,
				3,
				4
			},
			{
				8,
				6,
				1,
				2,
				3
			},
			{
				7,
				8,
				6,
				1,
				2
			},
			{
				9,
				7,
				8,
				6,
				1
			}
		};
		MyMatrix obj = new MyMatrix(matrix);
		obj.is_toeplitz(matrix);
	}
}

Output

Toeplitz matrix
<?php
/*
  Php Program
  Find if given matrix is Toeplitz or not
*/
class MyMatrix {
	public $rows;
	public $cols;

	function __construct($matrix) {
		//Get matrix size
		$this->rows = count($matrix);
		$this->cols = count($matrix[0]);
	}
	//Function which is check diagonal element values are similar or not

	public 	function diagonal($matrix, $start_row, $start_col) {
		$element = $matrix[$start_row][$start_col];
		for ($i = $start_row + 1, $j = $start_col + 1; $i < $this->rows && $j < $this->cols; ++$i, ++$j) {
			if ($matrix[$i][$j] != $element) {
				return false;
			}
		}
		return true;
	}
	// Determine whether matrix is Toeplitz matrix or not

	public 	function is_toeplitz($matrix) {
		$result = true;
		for ($i = 0; $i < $this->cols && $result == true; ++$i) {
			//check upper triangular
			$result = $this->diagonal($matrix, 0, $i);
		}
		for ($i = 1; $i < $this->rows && $result == true; ++$i) {
			//check lower triangular
			$result = $this->diagonal($matrix, $i, 0);
		}
		if ($result == true) {
			echo("Toeplitz matrix\n");
		} else {
			echo("Not a Toeplitz matrix\n");
		}
	}
}

function main() {
	//Define matrix element
	$matrix = array(array(1, 2, 3, 4, 5), array(6, 1, 2, 3, 4), array(8, 6, 1, 2, 3), array(7, 8, 6, 1, 2), array(9, 7, 8, 6, 1));
	$obj = new MyMatrix($matrix);
	$obj->is_toeplitz($matrix);

}
main();

Output

Toeplitz matrix
/*
  Node Js Program
  Find if given matrix is Toeplitz or not
*/
class MyMatrix {
	
	constructor(matrix) {
		//Get matrix size
		this.rows = matrix.length;
		this.cols = matrix[0].length;
	}

	//Function which is check diagonal element values are similar or not
	diagonal(matrix, start_row, start_col) {
		var element = matrix[start_row][start_col];
		for (var i = start_row + 1, j = start_col + 1; i < this.rows && j < this.cols; ++i, ++j) {
			if (matrix[i][j] != element) {
				return false;
			}
		}

		return true;
	}

	// Determine whether matrix is Toeplitz matrix or not
	is_toeplitz(matrix) {
		var result = true;
		for (var i = 0; i < this.cols && result == true; ++i) {
			//check upper triangular
			result = this.diagonal(matrix, 0, i);
		}

		for (var i = 1; i < this.rows && result == true; ++i) {
			//check lower triangular
			result = this.diagonal(matrix, i, 0);
		}

		if (result == true) {
			process.stdout.write("Toeplitz matrix\n");
		} else {
			process.stdout.write("Not a Toeplitz matrix\n");
		}
	}
}

function main(args) {
	//Define matrix element
	var matrix = [
		[1, 2, 3, 4, 5],
		[6, 1, 2, 3, 4],
		[8, 6, 1, 2, 3],
		[7, 8, 6, 1, 2],
		[9, 7, 8, 6, 1]
	];
	var obj = new MyMatrix(matrix);
	obj.is_toeplitz(matrix);
}

main();

Output

Toeplitz matrix
#   Python 3 Program
#   Find if given matrix is Toeplitz or not
class MyMatrix :
	
	def __init__(self, matrix) :
		# Get matrix size
		self.rows = len(matrix)
		self.cols = len(matrix[0])
	
	# Function which is check diagonal element values are similar or not
	def diagonal(self, matrix, start_row, start_col) :
		element = matrix[start_row][start_col]
		i = start_row + 1
		j = start_col + 1
		while (i < self.rows and j < self.cols) :
			if (matrix[i][j] != element) :
				return False
			
			i += 1
			j += 1
		
		return True
	
	#  Determine whether matrix is Toeplitz matrix or not
	def is_toeplitz(self, matrix) :
		result = True
		i = 0
		while (i < self.cols and result == True) :
			# check upper triangular
			result = self.diagonal(matrix, 0, i)
			i += 1
		
		i = 1
		while (i < self.rows and result == True) :
			# check lower triangular
			result = self.diagonal(matrix, i, 0)
			i += 1
		
		if (result == True) :
			print("Toeplitz matrix\n", end = "")
		else :
			print("Not a Toeplitz matrix\n", end = "")
		
	

def main() :
	matrix = [
		[1, 2, 3, 4, 5],
		[6, 1, 2, 3, 4],
		[8, 6, 1, 2, 3],
		[7, 8, 6, 1, 2],
		[9, 7, 8, 6, 1]
	]
	obj = MyMatrix(matrix)
	obj.is_toeplitz(matrix)


if __name__ == "__main__":
	main()

Output

Toeplitz matrix
# Ruby Program
# Find if given matrix is Toeplitz or not
class MyMatrix 
	# Define the accessor and reader of class MyMatrix
    attr_reader :rows, :cols
    attr_accessor :rows, :cols
	def initialize(matrix) 
		# Get matrix size
		self.rows = matrix.length
		self.cols = matrix[0].length
	end
	# Function which is check diagonal element values are similar or not
	def diagonal(matrix, start_row, start_col) 
		element = matrix[start_row][start_col]
		i = start_row + 1
		j = start_col + 1
		while (i < self.rows && j < self.cols) 
			if (matrix[i][j] != element) 
				return false
			end
			i += 1
			j += 1
		end
		return true
	end
	#  Determine whether matrix is Toeplitz matrix or not
	def is_toeplitz(matrix) 
		result = true
		i = 0
		while (i < self.cols && result == true) 
			# check upper triangular
			result = self.diagonal(matrix, 0, i)
			i += 1
		end
		i = 1
		while (i < self.rows && result == true) 
			# check lower triangular
			result = self.diagonal(matrix, i, 0)
			i += 1
		end
		if (result == true) 
			print("Toeplitz matrix\n")
		else 
			print("Not a Toeplitz matrix\n")
		end
	end
end
def main() 
	matrix = [
		[1, 2, 3, 4, 5],
		[6, 1, 2, 3, 4],
		[8, 6, 1, 2, 3],
		[7, 8, 6, 1, 2],
		[9, 7, 8, 6, 1]
	]
	obj = MyMatrix.new(matrix)
	obj.is_toeplitz(matrix)
end


main()

Output

Toeplitz matrix

/*
  Scala Program
  Find if given matrix is Toeplitz or not
*/
class MyMatrix(var rows: Int,
		var cols: Int) {

	def this(matrix: Array[Array[Int]]) {
		//Get matrix size
		this(matrix.length,matrix(0).length);
	}
	//Function which is check diagonal element values are similar or not
	def diagonal(matrix: Array[Array[Int]], start_row: Int, start_col: Int): Boolean = {
		val element: Int = matrix(start_row)(start_col);
		var i: Int = start_row + 1;
		var j: Int = start_col + 1;
		while (i < this.rows && j < this.cols) {
			if (matrix(i)(j) != element) {
				return false;
			}
			i += 1;
			j += 1;
		}
		return true;
	}
	// Determine whether matrix is Toeplitz matrix or not
	def is_toeplitz(matrix: Array[Array[Int]]): Unit = {
		var result: Boolean = true;
		var i: Int = 0;
		while (i < this.cols && result == true) {
			//check upper triangular
			result = this.diagonal(matrix, 0, i);
			i += 1;
		}
		i = 1;
		while (i < this.rows && result == true) {
			//check lower triangular
			result = this.diagonal(matrix, i, 0);
			i += 1;
		}
		if (result == true) {
			print("Toeplitz matrix\n");
		} else {
			print("Not a Toeplitz matrix\n");
		}
	}
}
object Main {
	def main(args: Array[String]): Unit = {
		val matrix: Array[Array[Int]] = Array(
			Array(1, 2, 3, 4, 5),
			Array(6, 1, 2, 3, 4),
			Array(8, 6, 1, 2, 3),
			Array(7, 8, 6, 1, 2),
			Array(9, 7, 8, 6, 1));
		val obj: MyMatrix = new MyMatrix(matrix);
		obj.is_toeplitz(matrix);
	}
}

Output

Toeplitz matrix
/*
  Swift Program
  Find if given matrix is Toeplitz or not
*/
class MyMatrix {
	var rows: Int;
	var cols: Int;
	init(_ matrix: [
		[Int]
	]) {
		//Get matrix size
		self.rows = matrix.count;
		self.cols = matrix[0].count;
	}
	//Function which is check diagonal element values are similar or not
	func diagonal(_ matrix: [
		[Int]
	], _ start_row: Int, _ start_col: Int) -> Bool {
		let element: Int = matrix[start_row][start_col];
		var i: Int = start_row + 1;
		var j: Int = start_col + 1;
		while (i < self.rows && j < self.cols) {
			if (matrix[i][j] != element) {
				return false;
			}
			i += 1;
			j += 1;
		}
		return true;
	}
	// Determine whether matrix is Toeplitz matrix or not
	func is_toeplitz(_ matrix: [
		[Int]
	]) {
		var result: Bool = true;
		var i: Int = 0;
		while (i < self.cols && result == true) {
			//check upper triangular
			result = self.diagonal(matrix, 0, i);
			i += 1;
		}
		i = 1;
		while (i < self.rows && result == true) {
			//check lower triangular
			result = self.diagonal(matrix, i, 0);
			i += 1;
		}
		if (result == true) {
			print("Toeplitz matrix\n", terminator: "");
		} else {
			print("Not a Toeplitz matrix\n", terminator: "");
		}
	}
}
func main() {
	let matrix: [
		[Int]
	] = [
		[1, 2, 3, 4, 5],
		[6, 1, 2, 3, 4],
		[8, 6, 1, 2, 3],
		[7, 8, 6, 1, 2],
		[9, 7, 8, 6, 1]
	];
	let obj: MyMatrix = MyMatrix(matrix);
	obj.is_toeplitz(matrix);
}
main();

Output

Toeplitz matrix
/*
  Java Program
  Find if given matrix is Toeplitz or not
*/
public class MyMatrix {

  public int rows;
  public int cols;
  public MyMatrix(int [][]matrix)
  {
    //Get matrix size
    this.rows = matrix.length;
    this.cols = matrix[0].length;

  }
  //Function which is check diagonal element values are similar or not
  public boolean diagonal(int [][]matrix,int start_row,int start_col)
  {
    int element = matrix[start_row][start_col];
    for (int i = start_row+1,j=start_col+1; i < this.rows && j<this.cols; ++i,++j)
    {
      if(matrix[i][j]!=element)
      {
        return false;
      }
    }
   return true;
  }
  // Determine whether matrix is Toeplitz matrix or not
  public void is_toeplitz(int [][]matrix)
  {
    
    boolean result=true;

    for (int i = 0; i < this.cols && result==true; ++i)
    {
      //check upper triangular
      result=diagonal(matrix,0,i);
    }
    for (int i = 1; i < this.rows && result==true; ++i)
    {
      //check lower triangular
      result=diagonal(matrix,i,0);
    }

    if(result==true)
    {
      System.out.print("Toeplitz matrix\n");
    }
    else
    {
      System.out.print("Not a Toeplitz matrix\n");
    }
   
  }

  public static void main(String[] args) 
  {
    //Define matrix element
    int [][]matrix ={

      {1,  2,  3,  4,  5},
      {6,  1,  2,  3,  4}, 
      {8,  6,  1,  2,  3},
      {7,  8,  6,  1,  2},
      {9,  7,  8,  6,  1}
    }; 
    MyMatrix obj = new MyMatrix(matrix);
    obj.is_toeplitz(matrix);
  }
}

Output

Toeplitz matrix

Output Explanation

The provided C code implements the above algorithm to determine whether a given matrix is a Toeplitz matrix. It iterates through the matrix and checks the diagonal elements starting from each element in the first row and the first column. If all diagonals have the same elements, it outputs "Toeplitz matrix." Otherwise, it outputs "Not a Toeplitz matrix."

Time Complexity

The time complexity of the algorithm is O(ROW * COL), where ROW is the number of rows and COL is the number of columns in the matrix. This is because the algorithm iterates through each element of the matrix and checks the diagonal elements for each element.

Last updated on by Kalkicode




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