Exercise 2.2.6

Answers

We want to prove Ha = r, let’s prove that Ha − r = 0, we have

Ha − r = a − 2(a − r)(a − r)T (a − r)T(a − r)a − r = (a − r) − 2(a − r)(a − r)T (a − r)T(a − r)a = (a − r)(a − r)T(a − r) − 2(a − r)(a − r)Ta (a − r)T(a − r) = (a − r)(a − r)T(a + r) (a − r)T(a − r)

We only need to prove that (a − r)(a − r)T(a + r) = 0. Expand the products and apply aTa = rTr, we have

(a − r)(a − r)T(a + r) = (aaT − arT − raT + rrT)(a + r) = aaTa − arTa − raTa + rrTa + aaTr − arTr − raTr + rrTr = rrTa − arTa + aaTr − raTr = (a − r)(aTr − rTa) = 0

The second to last step, we have used aTr = rTa since it’s scalar.

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2020-03-20 00:00
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