Spectral Analysis of a Square Matrix

In linear algebra and spectral theory, eigenvalues are fundamental scalar quantities that characterize the behavior of linear transformations. For a square matrix A, an eigenvalue λ is a scalar such that there exists a non-zero vector v (called an eigenvector) satisfying:

$$A \cdot v = \lambda \cdot v$$

This equation reveals that when the matrix A acts on its eigenvector v, the result is simply a scaled version of v by the factor λ. Eigenvalues capture the "stretching factors" of the transformation along special directions.

The Characteristic Polynomial Approach:

For a 2×2 matrix of the form:

$$A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$$

The eigenvalues can be found by solving the characteristic equation:

$$\det(A - \lambda I) = 0$$

This expands to the quadratic equation:

$$\lambda^2 - \text{trace}(A) \cdot \lambda + \det(A) = 0$$

Where:

• trace(A) = a + d (sum of diagonal elements)
• det(A) = ad - bc (determinant of the matrix)

Using the quadratic formula, the two eigenvalues are:

$$\lambda_{1,2} = \frac{\text{trace}(A) \pm \sqrt{\text{trace}(A)^2 - 4 \cdot \det(A)}}{2}$$

Your Task: Write a Python function that computes the eigenvalues of a 2×2 matrix. The function should:

• Apply the characteristic polynomial method to derive the eigenvalues
• Return a list containing both eigenvalues, sorted from highest to lowest
• Handle cases where eigenvalues may be equal (repeated eigenvalues)

matrix = [[2, 1], [1, 2]]

[3.0, 1.0]

matrix = [[4, 0], [0, 2]]

[4.0, 2.0]

matrix = [[1, 0], [0, 1]]

[1.0, 1.0]