Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Digital systems, linear algebra, introduction to formal proof / 5 / c

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- Solution c

Solution

Using the unit length of

q_{i}

,

P_{i}^{2} = q_{i} q_{i}^{T} q_{i} q_{i}^{T} = q_{i} (q_{i}^{T} q_{i}) q_{i}^{T} = q_{i} q_{i}^{T} = P_{i} .

Then ordinary matrix expansion gives

(I_{n} - P_{i})^{2} = I_{n} - 2 P_{i} + P_{i}^{2} = I_{n} - P_{i}

. Both $P_{i}$ and its complementary matrix are projectors. The second calculation in fact holds for every idempotent matrix, without requiring the matrix to equal its transpose.

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