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:
- $A_1 = v$
- $A_2 = v^t$
- $A_3 = v^tw$
- $A_4 = vw^t$
- $A_5 = I - 2vv^t$
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!