## Householder Matrix

Hi all,..I need help for this problem

Let $A \in \mathbb{C}^{n \times n}$.
Let $S=\left[
\begin{array}{cc}
I & 0 \\
0 & H%
\end{array}%
\right] \in
%TCIMACRO{\U{2102} }%
�ginExpansion
\mathbb{C}
%EndExpansion
^{n \times n}$
, where $H$ is a $k \times k$ Householder matrix such that $SA$ has zero component in the first column, from line $n - k + 2$ to $n$.

Consider matrix $B = SAS^{*} = SAS$.
Check whether or not $B$ has zero component in the first column, from line $n - k + 2$ to $n$.

I am still confused about the householder matrix $H$, is it a tridiagonal matrix? what are the properties?