Trying to help deriving a special case of Sherman Morrison formula, I found something peculiar.
I made the ansatz $$(I+cd^T)^{-1} = I + A$$
Which lead me to two different ways to build identity matrix: $$(I+A)(I+cd^T) = (I+cd^T)(I+A)$$
$$I+cd^T+A+Acd^T = I + A + cd^T + cd^TA$$
$$Acd^T = cd^TA$$
Now since $cd^T$ is a rank 1 outer product, it cannot have an inverse, but $A$ might:
$$cd^T = A^{-1}(cd^T)A$$
$$A(cd^T)A^{-1} = cd^T$$
In other words the rank-1 update $cd^T$ is similar to itself under $A$ transformation.
What does this lead to? Is it common to have this relationship between matrices, or is it something special?
