matrix structures obtainable by multiplication by unitary matrices Out of "Numerical Linear Algebra" by Trefethen and Bau:


  Suppose we have $3\times 3$ matrices and wish to introduce zeroes by left- and/or right-multiplications by unitary matrices $Q_j$ such as Householder reflectors or Givens rotations. Consider the following matrix structures:
  $$
(a)\begin{bmatrix}
\times &amp; \times &amp; 0\\
0 &amp; \times &amp; \times\\
0 &amp; 0 &amp; \times\\
\end{bmatrix}\ \ \ \ \ \ \ \ (b)\begin{bmatrix}
\times &amp; \times &amp; 0\\
\times &amp; 0 &amp; \times\\
0 &amp; \times &amp; \times\\
\end{bmatrix}\ \ \ \ \ \ \ \ (c)\begin{bmatrix}
\times &amp; \times &amp; 0\\
0 &amp; 0 &amp; \times\\
0 &amp; 0 &amp; \times\\
\end{bmatrix}
$$
  
  (where $\times$ "represents an entry that is not necessarily zero")
  
  For each one, decide which of the following situations olds and justify your claim
  
  (i) Can be obtained by a sequence of left-multiplications by matrices $Q_j$
  
  (ii) Not (i), but can be obtained by a sequence of left- and right-multiplication by matrices $Q_j$
  
  (iii) Cannot be obtained by any sequence of left- and right-multiplications by matrices $Q_j$.


Not sure how to argue, here, and not sure how $Q_j$'s being unitary factors in.

Some help would be hot.
Hints.

(a) Can every rank-1 matrix be reduced to this form?

(b) Handle the first column by Householder reflection. Then try to handle the $2\times2$ submatrix at the top right corner by singular value decomposition. Note that in such a decomposition $USV^\ast$, the unitary matrices $U,V$ may have determinants $-1$. How do you remedy for this?

(c) What is the rank of a matrix in this pattern?
Hints.

(a) Can every rank-1 matrix be reduced to this form?

(b) Handle the first column by Householder reflection. Then try to handle the $2\times2$ submatrix at the top right corner by singular value decomposition. Note that in such a decomposition $USV^\ast$, the unitary matrices $U,V$ may have determinants $-1$. How do you remedy for this?

(c) What is the rank of a matrix in this pattern?