Show that $(I-\mathbf x\mathbf y^T)^{-1} = I-\frac{1}{\mathbf x^T\mathbf y- 1}\mathbf x\mathbf y^T$

Question

Let $x,y\in\mathbb R^n$ and suppose that $x^Ty \neq 1$. Show that $(I-\mathbf x\mathbf y^T)^{-1} = I-\frac{1}{\mathbf x^T\mathbf y- 1}\mathbf x\mathbf y^T$. Note, I need to compute this directly not as some special case.

We need to show that two things:

1. $(I-x y^T)(I-\frac{1}{x^T y- 1} x y^T) = I$, and

2. $(I-\frac{1}{x^T y- 1} x y^T)(I-x y^T) = I$

By definition.

For (1) I have (EDIT 1) $$\begin{equation}\begin{split}(I-x y^T)(I-\frac{1}{x^T y- 1} x y^T) &= I - \frac{1}{x^T y- 1} x y^T-x y^T + \frac{1}{x^T y- 1} x y^Tx y^T \\ &=I - \frac{1}{x^T y- 1} x y^T-x y^T\left(\frac{x^T y- 1}{x^T y- 1}\right) + \frac{1}{x^T y- 1} x y^Tx y^T \\ &= I - \frac{1}{x^T y- 1} \left[x y^T + (x^Ty-1)x y^T - x y^Txy^T\right] \\ &=I - \frac{1}{x^T y- 1} \left[x y^T + x^Tyx y^T-x y^T -x y^Txy^T\right] \\ &=I - \frac{1}{x^T y- 1}\left[x^Tyx y^T -x y^Txy^T\right] \\ &=I - \frac{1}{x^T y- 1}\left[x^Tyx y^T -x (y^Tx)y^T\right] \\ &=I - \frac{1}{x^T y- 1}\left[x^Tyx y^T -(y^Tx)x y^T\right]\\ &=I - \frac{1}{x^T y- 1}\left[x^Tyx y^T -x^Tyx y^T\right] \\ &= I.\end{split}\end{equation}$$

With the help in the comments! I think I got it.

And similar multiplication for (2)...

Answer 1

For (1), we will use the fact that $\frac{x^Ty-1}{x^Ty-1} = 1$ and $x^Ty = y^Tx$. We also use the fact that $xy^Txy^T = (x(y^Tx)y^T) = (y^Tx)xy^T =y^Txxy^T =x^Tyxy^T $. Obser for (1) we have that $$\begin{equation}\begin{split}\left(I-x y^T\right)\left(I-\frac{1}{x^T y- 1} x y^T\right) &= I - \frac{1}{x^T y- 1} x y^T-x y^T + \frac{1}{x^T y- 1} x y^Tx y^T \\ &=I - \frac{1}{x^T y- 1} x y^T-x y^T\left(\frac{x^T y- 1}{x^T y- 1}\right) + \frac{1}{x^T y- 1} x y^Tx y^T \\ &= I - \frac{1}{x^T y- 1} \left[x y^T + (x^Ty-1)x y^T - x y^Txy^T\right] \\ &=I - \frac{1}{x^T y- 1} \left[x y^T + x^Tyx y^T-x y^T -x^Tyxy^T\right] \\ &=I - \frac{1}{x^T y- 1}\left[x^Tyx y^T -x^Tyxy^T\right] \\ &=I.\end{split}\end{equation}$$ For (2) we have $$\begin{equation}\begin{split}\left(I-\frac{1}{x^T y- 1} x y^T\right)\left(I-x y^T\right) &= I - xy^T - \frac{1}{x^T y- 1} x y^T+ \frac{1}{x^T y- 1} x y^Txy^T \\ &= I. \text{ by (1) calculation}\end{split}\end{equation}$$

Thus, $(I-x y^T)^{-1} = I-\frac{1}{x^T y- 1} x y^T$.