Hi all,..I need help for this problem

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

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


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