Eigenvalues of a 2×2 Matrix
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Example: the covariance matrix of two variables, each with variance 2 and a covariance of 1 between them. In which directions does it only stretch, and by how much? Change the entries of the matrix A = (a, b; c, d) to work out your own case.
Step by step
What each term means
- Eigenvalue (λ)
- A stretch factor: there is a direction v in which the matrix only multiplies by λ, A v = λ v. A 2×2 matrix has two, real or complex.
- Trace (tr A)
- The sum of the diagonal, a + d. It is also the sum of the two eigenvalues.
- Root of the Discriminant
- With (tr A)² − 4 det A positive, two distinct real eigenvalues; zero, a repeated one; negative, complex ones — the matrix rotates the plane. det A = ad − bc is the product of the eigenvalues.
- Divided by 2
- It is the quadratic formula applied to λ² − (tr A) λ + det A = 0, the characteristic polynomial.