Past exam of the options I course of the University of Oxford / 2004 / F / F1 / b

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- Solution b

Solution

The requested base point is the identity matrix

I

, not the zero matrix. Since

I

commutes with every

H

,

exp (I + H) = exp (I) exp (H) = e [I + H + O (∥ H ∥^{2})] .

The matrix power series justifies the quadratic remainder near zero. Consequently

D exp_{I} (H) = eH .

At zero the derivative would instead be

H

; replacing the printed base point would lose the scalar factor.

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