Theorems exp

Question 1

For all A ∈ C^n×n and all t ∈ R,

d/dt e^(At) =

Answer 1

Ae^(At) = e^(At)A

Question 2

For A, B ∈ C^n×n, e^ ((A+B)t) = e^(At)e^(Bt) for all t if and only if

Answer 2

AB = BA

Question 3

Jordan form

Answer 3

Let A= XJX^-1 ∈ C^n×n with Jordan canonical form
J = diag(J1, J2, . . . , Jp),
where Ji is an mi × mi Jordan block with eigenvalue λi. Then
e^(At) = Xe^(Jt)X^-1
=X diag(e^(J1t), e^(J2t),…e^(Jpt)) X^-1

where e^(Jit) = e^(λit) [1 t t^2/2!….t^(mi-1)/(mi-1)!

                                                 .....t
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