Lecture 151:- Toeplitz Matrix

A Toeplitz matrix is a special type of matrix where each descending diagonal from left to right is constant. In other words, all the elements on any given diagonal in a Toeplitz matrix are equal. The term "Toeplitz" is often used to refer to matrices with this property, regardless of whether they are square or rectangular.

Here is an example of a Toeplitz matrix:

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1 2 3 4 5 1 2 3 6 5 1 2 7 6 5 1

In this matrix, each descending diagonal from left to right is constant: [1], [5, 2], [6, 1, 3], [7, 5, 2, 4].

Toeplitz matrices have several interesting properties, and they are used in various applications, such as signal processing, image processing, and solving linear equations.

Let's create a C++ program to check whether a given matrix is a Toeplitz matrix or not:

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#include <iostream> #include <vector> bool isToeplitzMatrix(std::vector<std::vector<int>>& matrix) {    int rows = matrix.size();    int cols = matrix[0].size();    // Check descending diagonals in each row    for (int i = 0; i < rows - 1; i++) {        for (int j = 0; j < cols - 1; j++) {            if (matrix[i][j] != matrix[i + 1][j + 1]) {                return false;            }        }    }    return true; } int main() {    std::vector<std::vector<int>> matrix = {        {1, 2, 3, 4},        {5, 1, 2, 3},        {6, 5, 1, 2},        {7, 6, 5, 1}    };    if (isToeplitzMatrix(matrix)) {        std::cout << "The given matrix is a Toeplitz matrix." << std::endl;    } else {        std::cout << "The given matrix is not a Toeplitz matrix." << std::endl;    }    return 0; }

Output:

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The given matrix is a Toeplitz matrix.

In this program, we define the function isToeplitzMatrix that takes a 2D vector matrix as input and checks whether it is a Toeplitz matrix. The function iterates through each descending diagonal in each row (excluding the last row and last column) and compares the elements to check if they are all equal. If any diagonal is not constant, the function returns false, indicating that the matrix is not a Toeplitz matrix. Otherwise, it returns true.

You can use this function to check whether a given matrix is a Toeplitz matrix or not.