How to derived the inverse matrix I highlined? I tried to search for the inverse matrix formula and then apply the adjoint formula to solve it but had no idea how to start let alone continue.
2026-03-31 14:24:26.1774967066
An inverse matrix problem on a statistics book
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Notice that \begin{align}(I+vv^T)^2&=I+2vv^T+vv^Tvv^T=I+2vv^T+(v^Tv)vv^T=I+(2+v^Tv)vv^T=\\&= I-(2+v^Tv)I+(2+v^Tv)(I+vv^T)\\ (1+v^Tv)I&=(2+v^Tv)(I+vv^T)-(I+vv^T)^2\\I&=\left(I-\frac1{1+v^Tv}vv^T\right)(I+vv^T)\end{align}
The idea is basically that every matrix $A$ of rank $1$ satisfies a polynomial of degree two, and this introduces relations between $(\beta I+A)^2$ and $\beta I+A$ and $I$. This results in an equation $\alpha_1I=\alpha_2(\beta I+A)+\alpha_3(\beta I+A)^2$.