Matrix exponential: e^A = I + A + A²/2! + A³/3! + ... For 2×2 with distinct eigenvalues λ₁,λ₂: e^(At) = (e^(λ₁t)(A−λ₂I) − e^(λ₂t)(A−λ₁I))/(λ₁−λ₂). Solution to dx/dt = Ax is x(t) = e^(At)x(0).
How to Use
Enter a 2×2 matrix:
- A: System matrix
- t: Time parameter
- Output: e^(At)
Computation Methods
- Distinct real λ: direct formula
- Repeated λ: e^(λt)(I + (A−λI)t)
- Complex λ=a±bi: e^(at)(cos(bt)I + sin(bt)(A−aI)/b)
ODE Connection
dx/dt = Ax has solution x(t) = e^(At)x₀. For forced systems dx/dt = Ax + f(t): x(t) = e^(At)x₀ + ∫₀ᵗ e^(A(t−s))f(s)ds.
Step-by-Step Instructions
- 1Enter matrix A.
- 2Set time t.
- 3View e^(At).
- 4Get eigenvalues.
- 5Solve ODE.