In a published paper I saw the following
$$\log \left(\mathbf{I} + \mathbf{T}\mathbf {Hpp^HH^H}\right)= \log(1+\mathbf {p^HH^HTHp})$$
where uppercase means a matrix while lower case means vector and $()^H$ means cojugate transpose of matrix and this was justified by the following
$$|\mathbf{I+XY}| = |\mathbf{I+YX}|$$
where $\mathbf{X}=\mathbf{T}\mathbf {Hp}$ while $\mathbf{Y}=\mathbf{p^HH^H}$
I have a slightly different problem
$$\log \left(\mathbf{I} + \mathbf{T}\mathbf {w^HHH^Hw}\right)=???$$ and I am trying replace the argument by scalar as the authors of paper did.
Looking forward for help