Proof: The differentiability and domain assumptions are sufficient to define the primary matrix functions f(tA) (for t in a neighborhood of t0) and f′(t0A). Let A have Jordan canonical form A = SJS−1, where J has the form (6.2.5). Then
so it suffices to prove (6.2.35) in the special case of a single Jordan block.
Let Jk(λ) be one of the Jordan blocks in the Jordan canonical form of A, so f(t) is at least k-times differentiable at t = t0λ and is at least (k − 1)-times differentiable in a neighborhood of t0λ. For t ≠ 0, let St ≡ diag(1, 1/t