Module matrix_exp 

The exact matrix exponential e^{t·M} of a rational matrix.

The characteristic polynomial χ of M annihilates it, so e^{t·M} = r(M) for the polynomial r of degree below n that agrees with e^{t·x} on the roots of χ to their multiplicities: r^{(j)}(λ) = t^j·e^{t·λ} for j < m(λ) (Hermite interpolation). Writing r = Σ c_k·x^k, those conditions are a confluent Vandermonde system V·c = (t^j·e^{t·λ}), so c = V⁻¹·(t^j·e^{t·λ}) with V⁻¹ exact over the complex algebraic numbers, and each entry of e^{t·M} is a sum Σ_λ β_λ·e^{t·λ}. A conjugate pair of eigenvalues a ± b·i contributes 2·e^{t·a}·(Re β·cos(t·b) − Im β·sin(t·b)), so every entry is a real number in the exponential exact form: e^{[[0, 1], [−1, 0]]} is [[cos(1), sin(1)], [−sin(1), cos(1)]] exactly.

Functions

characteristic_polynomial

det(x·I − M) by Faddeev–LeVerrier; None for a matrix that is not square.

exp_matrix

e^{t·M}, every entry exact; None for a matrix that is not square.