Toeplitz Matrix

Python solution:

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class Solution:
    def isToeplitzMatrix(self, matrix: List[List[int]]) -> bool:
        # Step 1: Iterate over the elements in the matrix, skipping the last row and the last column
        for row in range(len(matrix) - 1):
            for col in range(len(matrix[0]) - 1):
                # Step 2: Check if current element is not equal to the element at the next row and column
                if matrix[row][col] != matrix[row + 1][col + 1]:
                    # If they are not equal, return False
                    return False

        # Step 3: If we get through the entire matrix without finding unequal diagonal elements, return True
        return True

This function iterates through the matrix, checking each element and the element diagonally down and to the right of it. If these elements are not equal, the matrix is not a Toeplitz matrix, and we return False. If we get through the entire matrix without finding unequal diagonal elements, the matrix is a Toeplitz matrix, and we return True.

Identifying Problem Isomorphism

“Toeplitz Matrix” can be approximately mapped to “Diagonal Traverse”.

Reasoning:

Both problems involve a non-standard traversal of a 2D matrix, specifically along the diagonals, though the specific operations and patterns differ.

  1. “Toeplitz Matrix” is simpler, it requires you to traverse the matrix diagonally and check if each diagonal from top-left to bottom-right has the same elements.

  2. “Diagonal Traverse”, on the other hand, is more complex. It requires you to traverse the elements of the matrix diagonally, alternating between upward and downward directions, and return the elements as a 1D array.

These are approximately isomorphic. They share a similar requirement of traversing a matrix in a non-standard, diagonal pattern. If you have a grasp of how to manipulate indices to traverse diagonally, both problems should be approachable.

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def isToeplitzMatrix(self, matrix: List[List[int]]) -> bool:
        for i in range(1,len(matrix)):
            for j in range(1,len(matrix[0])):
                if matrix[i-1][j-1] != matrix[i][j]:
                    return False
        return True

Problem Classification

This belongs to the domain of “Linear Algebra”, more specifically it falls under “Matrix operations”. The task involves a particular property checking operation on a 2D matrix.

‘What’ Components:

  1. An m x n matrix: This is the primary input. The solution will need to process this matrix to determine if it adheres to the required property.
  2. Toeplitz Matrix: This is a specific property of the matrix that needs to be checked. A Toeplitz matrix is one in which every diagonal from top-left to bottom-right has the same elements.
  3. Boolean result: The problem requires a yes/no (true/false) result indicating if the given matrix is a Toeplitz matrix or not.

This is a “Property Checking” problem, where we need to check if a given input (in this case, a matrix) satisfies a specific property. The property is defined by the mathematical concept of a Toeplitz matrix. The main task is to verify this property across all relevant elements of the matrix (in this case, across all diagonals from top-left to bottom-right).

Language Agnostic Coding Drills

  1. Dissect the Code and Identify Each Distinct Concept:

Concept 1: Basic Function Definition - This is the primary structure of defining a function in Python.

Concept 2: Iterating over a 2D array - The nested for-loops in the code iterate over a 2D matrix.

Concept 3: Indexing in a 2D array - Retrieving the elements of the 2D matrix using indices.

Concept 4: Conditional Statement - Using the ‘if’ condition to check the equality of elements.

Concept 5: Early return in a function - The function returns False as soon as it detects the matrix isn’t a Toeplitz.

Concept 6: Diagonal traversal in a 2D array - By accessing elements at (i-1, j-1) and (i, j), the code is checking diagonally consecutive elements.

  1. Increasing Difficulty and Description:

Level 1: Basic Function Definition - This is the foundation of any code; defining a function to encapsulate a piece of logic.

Level 2: Conditional Statement - Building on the basic structure, using ‘if’ to add decision-making capability.

Level 3: Iterating over a 2D array - Requires understanding of loops and 2D data structures.

Level 4: Indexing in a 2D array - Building on iteration, this concept requires understanding how to retrieve elements from a 2D structure.

Level 5: Early return in a function - Requires understanding the flow of control in a function.

Level 6: Diagonal traversal in a 2D array - A more complex form of iteration that requires a clear mental model of the matrix.

  1. Problem-solving approach:

a. Understanding the Problem Statement: The problem requires us to check if a given 2D matrix is Toeplitz, which means all its diagonals from top-left to bottom-right must have the same elements.

b. Conceptualizing the Approach: This involves iterating over the 2D matrix and for each cell, comparing it with the previous row and column cell (essentially checking the diagonal).

c. Building the Solution: Start by defining a function. Initialize two nested loops to traverse the 2D matrix. For each element, check if it’s equal to the element diagonally above it. If not, the matrix is not a Toeplitz matrix, so return False. If we go through the whole matrix without finding unequal diagonal elements, return True.

Targeted Drills in Python

  1. Coding Drills:

Concept 1: Basic Function Definition

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def greet():
    print("Hello, World!")

Concept 2: Iterating over a 2D Array

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matrix = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
for row in matrix:
    for elem in row:
        print(elem)

Concept 3: Indexing in a 2D Array

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matrix = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
print(matrix[1][2])  # prints 6

Concept 4: Conditional Statement

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num = 5
if num > 0:
    print("Positive number")

Concept 5: Early Return in a Function

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def is_positive(num):
    if num > 0:
        return True
    return False

Concept 6: Diagonal Traversal in a 2D Array

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matrix = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]
for i in range(len(matrix)):
    print(matrix[i][i])  # prints diagonal elements 1, 5, 9
  1. Problem-Specific Drills:

A problem-specific drill for this problem would involve the comparison of elements along the diagonals of a matrix. This concept is crucial for this problem as a Toeplitz matrix is defined by the property that all diagonals from top-left to bottom-right have the same element.

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matrix = [[1, 2], [3, 1], [4, 3]]
for i in range(1, len(matrix)):
    for j in range(1, len(matrix[0])):
        if matrix[i][j] != matrix[i-1][j-1]:
            print("Not a Toeplitz matrix")
  1. Integrating the Drills:

We start by defining a function that takes a 2D matrix as input. Within this function, we use two nested loops to iterate over the matrix. For each element, we use a conditional statement to check if it is equal to the element in the previous row and column. If we find an element that does not satisfy this condition, we know the matrix is not Toeplitz, and we return False immediately (early return). If we make it through the entire matrix without returning, that means all diagonals have the same elements, so we return True, indicating the matrix is Toeplitz. These drills come together to form the full solution.