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Exercise 3.3.5

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${(AP)}_{ij} = \sum A_{ik} P_{kj} = A_{ii} P_{ij} = x_{i} \frac{s_{i} + s_{j}}{x_{i} + x_{j}}$ and ${(PA)}_{ij} = \sum P_{ik} A_{kj} = P_{ij} A_{jj} = x_{j} \frac{s_{i} + s_{j}}{x_{i} + x_{j}}$ , so we have

$C = AP - P (- A) = AP + PA$ and its elements are $C_{ij} = {(AP)}_{ij} + {(PA)}_{ij} = \frac{(x_{i} + x_{j}) (s_{i} + s_{j})}{x_{i} + x_{j}} = s_{i} + s_{j}$

In the end $C = s 1^{T} + 1 s^{T}$ , which has rank-2.

niuers

2020-03-20 00:00

Exercise 3.3.5

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