To get the inverse of
, one can implement
Sherman-Morrison formula to transform the inverse matrix as
(13). Whereas
and
are complex vectors. Here, we drop the
subscript of vectors for a concise proof:
where
denotes the identity matrix,
is the
complex column vector,
is the transpose of
,
is the conjugation of
.
Therefore,
is a complex matrix, and
is constant. Therefore, we prove that
Sherman-Morrison formula can be applied in the complex space.