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Exercise 3.2.2

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Let $A = S + u u^{T}$ and $B = A + v v^{T}$ , also, let the eigenvalues of $A$ be $z_{1} \geq z_{2} \geq \dots \geq z_{n}$ .

So we have $z_{1} \geq λ_{1} \geq z_{2} \geq \dots \geq z_{n} \geq λ_{n}$ . We also have

$α_{1} \geq z_{1} \geq α_{2} \geq z_{2} \geq \dots \geq α_{n} \geq z_{n}$ .

For $2 \leq i \leq n$ , we have $α_{i - 1} \geq z_{i - 1} \geq λ_{i - 1} \geq z_{i} \geq α_{i + 1}$ , so in the end we have

$α_{1} \geq λ_{1} \geq α_{3} \geq λ_{3} \geq α_{5} \geq \dots λ_{n - 2} \geq α_{n} \geq λ_{n}$ and $α_{2} \geq λ_{2} \geq α_{4} \geq λ_{4} \geq α_{6} \geq \dots λ_{n - 2} \geq α_{n} \geq λ_{n}$

niuers

2020-03-20 00:00

Exercise 3.2.2

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