If A is an n*n matrix with n distinct eigenvalues λ_1,λ_2...,λ_k, then
prove that
n
e^(tA) = Σ e^(tλ_k) L_k(A)
k=1
where L_k(A) is a polynomial in A of degree n-1 given by the formula
n A - λ_jI
L_k(A) = Π(------------) for k=1,2,.....,n.
j=1 λ_k - λ_j
j≠k
n
[Hint: Consider f(t) = Σ e^(tλ_k) L_k(A) and show that f(t) = Af(t),
k=1
f(0)=I, where I is the identity. Conclude the proof using the ODE uniqueness
theorem.
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