# Toeplitz Matrix

## Description

A matrix is _Toeplitz_ if every diagonal from top-left to bottom-right has the same element.

Now given an `M x N` matrix, return `True` if and only if the matrix is _Toeplitz_.\
&#x20;

**Example 1:**

```
Input:
matrix = [
  [1,2,3,4],
  [5,1,2,3],
  [9,5,1,2]
]
Output: True
Explanation:
In the above grid, the diagonals are:
"[9]", "[5, 5]", "[1, 1, 1]", "[2, 2, 2]", "[3, 3]", "[4]".
In each diagonal all elements are the same, so the answer is True.
```

**Example 2:**

```
Input:
matrix = [
  [1,2],
  [2,2]
]
Output: False
Explanation:
The diagonal "[1, 2]" has different elements.
```

\
**Note:**

1. `matrix` will be a 2D array of integers.
2. `matrix` will have a number of rows and columns in range `[1, 20]`.
3. `matrix[i][j]` will be integers in range `[0, 99]`.

## Solutions

### My solution

```cpp
class Solution {
public:
    bool isToeplitzMatrix(vector<vector<int>>& matrix) {
        int m = matrix.size();
        int n = matrix[0].size();
        // line equations: j = i + k, k \in [1 - m, n - 1]
        for(int k = 1 - m; k < n; ++k){
            // i < m, j = i + k < n ==> i < min(m, n - k)
            for(int i = 1 + max(0, -k) ; i < min(m, n - k); ++i){
                if(matrix[i][i + k] != matrix[i - 1][i - 1 + k])
                    return false;
            }
        }
        return true;
    }
};
```

### A better solution

```cpp
class Solution {
public:
    bool isToeplitzMatrix(vector<vector<int>>& matrix) {
        int m = matrix.size();
        int n = matrix[0].size();
        for(int i = 1; i < m; ++i){
            for(int j = 1; j < n; ++j){
                if(matrix[i][j] != matrix[i - 1][j - 1])
                    return false;
            }
        }
        return true;
    }
};
```
