Eigenvalues of a 2x2 Matrix

Find the eigenvalues of a 2x2 matrix by solving the characteristic polynomial directly, without np.linalg.eig.

For

A = [[a, b],
     [c, d]]

the eigenvalues solve det(A - lambda I) = 0, which expands to

lambda^2 - (a + d) * lambda + (a * d - b * c) = 0

The coefficients are the trace and the determinant. Apply the quadratic formula with discriminant = trace^2 - 4 * det.

Assume the matrix has real eigenvalues, so the discriminant is not negative. Return the two eigenvalues in a list sorted in descending order. A repeated eigenvalue is returned twice.

Function signature

def eigenvalues_2x2(A):

Example

Input

eigenvalues_2x2(np.array([[2.0, 1.0], [1.0, 2.0]]))

Output

[3.0, 1.0]

Reasoning

The trace is 4 and the determinant is 3, so lambda^2 - 4 lambda + 3 = 0. The discriminant is 16 - 12 = 4, which gives (4 + 2) / 2 = 3 and (4 - 2) / 2 = 1.

Constraints and notes

• The discriminant is never negative in the tests.
• Return the values in descending order. A repeated value appears twice.