Eigenvalue Calculator

What are Eigenvalues?

In linear algebra, an eigenvector of a square matrix A is a non-zero vector v that, when multiplied by A, yields a scalar multiple of itself. That scalar multiple is called the eigenvalue, denoted by the Greek letter lambda (λ).

Av = λv

The Characteristic Equation

To find the eigenvalues of a 2x2 matrix, we solve the characteristic equation:

det(A - λI) = 0

For a 2x2 matrix:

This expands to the quadratic equation:

λ² - Tr(A)λ + Det(A) = 0

• Tr(A): The Trace of the matrix (a + d).
• Det(A): The Determinant of the matrix (ad - bc).

Real and Complex Roots

Since the characteristic equation is a quadratic, the eigenvalues are the roots. Depending on the discriminant (Δ = Tr² - 4Det), you can have:

• Distinct Real Roots: If Δ > 0.
• Repeated Real Root: If Δ = 0.
• Complex Conjugate Roots: If Δ < 0.