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?