Calculating pseudo inverse for specific vector products

Question

Let $v^tv = w^tw = 1$; $v,w \in\mathbb{R}^n$, and $I$ be the identity matrix. Calculate the pseudo inverse $A_k^{+}$ for:

1. $A_1 = v$
2. $A_2 = v^t$
3. $A_3 = v^tw$
4. $A_4 = vw^t$
5. $A_5 = I - 2vv^t$

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This is from a sample question for an exam I am going to have to take very soon, and even though I know how to calculate the pseudo inverse using the singular value decomposition (and for specific matrices that have full rank in one of their dimensions using special formulas), I am drawing completely blank here, I am not even sure how to approach this. I would appreciate any help!

Answer 1

You say you have formulas in the case when $A$ has full rank in one of their dimensions. The first three questions fall into this category.

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$A_4 A_4^+ A_4 = A_4$ can be written as $vw^t A_4^+ vw^t = vw^t$. If you can find a way to make $w^t A_4^+ v = 1$, that would work.

$A_4^+ = wv^t$

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Hint: Note that $A_5 A_5=A_5$.