Compute the **eigenvalues of a 2×2 matrix**, which describe how the matrix scales vectors along special directions called eigenvectors. You’ll implement a tiny helper that returns both eigenvalues in a consistent order.

The eigenvalues $\lambda$ and eigenvectors $v$ of a square matrix $A$ satisfy:
$$
A v = \lambda v
$$
The eigenvalues are found by solving the characteristic equation:
$$
\det(A - \lambda I) = 0
$$

The eigenvalues of
$$
A=\begin{bmatrix}a & b\\ c & d\end{bmatrix}
$$
are the roots of:
$$
\lambda^2 - (a+d)\lambda + (ad-bc)=0
$$

## Requirements

Implement the function
```python
def eigenvalues_2x2(A):
    \"""
    Computes the eigenvalues of a 2x2 matrix.

    Args:
        A (np.ndarray): A 2x2 matrix [[a, b], [c, d]].

    Returns:
        np.ndarray: An array [lambda1, lambda2] sorted in non-increasing order.
    \"""
    # Your solution below
```

**Rules:**
- Compute eigenvalues using the closed-form quadratic formula (not `np.linalg.eig`).
- Return a NumPy array of two floats, sorted from largest to smallest.
- If the eigenvalues are equal, return `[lambda, lambda]`.
- Use only NumPy and Python built-in libraries.
- Keep the implementation in this single function.

## Example

```python
A = np.array([
    [4.0, 2.0],
    [1.0, 3.0]
])

eigenvalues_2x2(A)
```

Output:
```python
array([5.0, 2.0])
```

Most Viewed Posts

Amazon-Only Product List

Eigenvalues

Solve this easy level ML coding question from General. The premier LeetCode for ML platform to master Python, SQL, and Machine Learning algorithms.

263. Eigenvalues

Examples

Constraints

Similar Questions