Directions a matrix does not turn
Multiply a vector by a matrix and, in general, it changes both length and direction. But for most matrices there are special directions where the vector only stretches or shrinks: A × v = λ × v. The vector v is an eigenvector and the stretch factor λ is its eigenvalue. For a 2×2 matrix, finding them takes nothing more than a quadratic equation.
The characteristic equation
For a matrix with rows (a, b) and (c, d):
λ² − (a + d) × λ + (ad − bc) = 0
The coefficient a + d is the trace and ad − bc the determinant. So:
- λ = (trace ± √(trace² − 4 × determinant)) ÷ 2.
- The sum of the eigenvalues equals the trace.
- Their product equals the determinant.
Worked example
Take the matrix with rows (4, 1) and (2, 3):
- Trace: 4 + 3 = 7. Determinant: 4 × 3 − 1 × 2 = 10.
- Discriminant: 7² − 4 × 10 = 9, positive, so there are two real eigenvalues.
- λ₁ = (7 + 3) ÷ 2 = 5 and λ₂ = (7 − 3) ÷ 2 = 2.
Check: 5 + 2 = 7 and 5 × 2 = 10.
Eigenvectors
For each eigenvalue, solve (A − λI) × v = 0. When b is not zero, the vector (b, λ − a) always works:
- For λ₁ = 5: (1, 5 − 4) = (1, 1). Indeed, A × (1, 1) = (5, 5) = 5 × (1, 1).
- For λ₂ = 2: (1, 2 − 4) = (1, −2). Indeed, A × (1, −2) = (2, −4) = 2 × (1, −2).
When b is zero, the calculator uses (λ − d, c) instead, and for a diagonal matrix the eigenvectors are simply the axes. Any nonzero multiple of an eigenvector is also an eigenvector, so (2, 2) or (−1, −1) are just as correct as (1, 1).
Three cases
| Discriminant | Eigenvalues | Example |
|---|---|---|
| Positive | Two different real values | rows (4, 1), (2, 3): 5 and 2 |
| Zero | One repeated real value | rows (2, 1), (0, 2): 2 twice |
| Negative | A complex conjugate pair | rows (0, −1), (1, 0): ± i |
A repeated eigenvalue may have only one independent eigenvector, as in the second example. A negative discriminant means the matrix includes a rotation: the rotation by 90° has eigenvalues 0 ± 1i, and no real direction stays fixed.
Why they matter
- Differential equations: for x’ = A x, solutions grow or decay like e^(λt), so negative real parts mean a stable system.
- Markov chains: the long-run distribution is the eigenvector with eigenvalue 1.
- Statistics: principal component analysis finds the eigenvectors of a covariance matrix; the eigenvalues give the variance along each one.
- Mechanics: the natural frequencies of vibrating systems come from eigenvalues.
For larger matrices, use the matrix calculator or numerical software, since the characteristic polynomial quickly becomes hard to solve by hand.