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 Incoming links: Past exam of the mathematical methods I course of the University of Oxford / 2013 / 11 / c / i

Setting

b = a

in i removes the cross product term and gives

σ (a)^{2} = (a \cdot a) I

. Repeated multiplication therefore yields, for nonnegative integers

m

σ (a)^{2 m} = (a \cdot a)^{m} I, σ (a)^{2 m + 1} = (a \cdot a)^{m} σ (a) .

The zeroth power is

I

. If negative powers are intended, they require

a \cdot a \neq = 0

, and the same parity formulas then follow from

σ (a)^{- 1} = σ (a) / (a \cdot a)

. Every nonzero real vector meets that condition; a nonzero complex null vector need not. For example,

a = (1, i, 0)

has

σ (a)^{2} = 0

and no inverse.

\hat{n}

, let

S = σ (\hat{n})

. Then

S^{2} = I

, so even powers are

I

and odd powers are

S

. Split the absolutely convergent matrix exponential into its even and odd series:

e^{i θS} = I m = 0 \sum \infty \frac{( - 1 ) ^{m} θ ^{2 m}}{( 2 m )!} + i S m = 0 \sum \infty \frac{( - 1 ) ^{m} θ ^{2 m + 1}}{( 2 m + 1 )!} = cos θ I + i sin θ σ (\hat{n}) .

The result holds for every real

θ

, not merely small angles. Its two scalar series are the ordinary cosine and sine series.