Understanding why the inverse looks like this.

Question

This a part from Carl D. Meyer book:

But I do not understand why the denominator in (3.8.1) contains $d^{T}c$ and not $c d^{T}$, could anyone explain this for me please?

Answer 1

Notice that $d^Tc$ is a scalar and $cd^T$ is a $n\times n$ matrix.
Using the textbook hint, multiply the matrix and its inverse to get $I$:

$(I+cd^T)(I - \dfrac{cd^T}{1+d^Tc}) =I+cd^T - \dfrac{cd^T+c(d^Tc)d^T}{1+d^Tc} = I+cd^T - \dfrac{cd^T(1+d^Tc)}{1+d^Tc}=I $

Answer 2

Woodbury matrix identity reads $$ \left(I + UV \right)^{-1} = I - U \left(I + VU \right)^{-1}V. $$ With $U = c$ and $V = d^T$, you get your answer. In particular $$ \left(I + VU \right)^{-1} = \frac1{1 + d^Tc}. $$