Characteristic polynomial of a $7 \times 7$ matrix whose entries are $5$ Avoiding too many steps, what is the characteristic polynomial of the following $7 \times 7$ matrix? And why?
\begin{pmatrix}5&5&5&5&5&5&5\\5&5&5&5&5&5&5\\5&5&5&5&5&5&5\\5&5&5&5&5&5&5\\5&5&5&5&5&5&5\\5&5&5&5&5&5&5\\5&5&5&5&5&5&5\end{pmatrix}

Here is a matrix $P,$ the columns are eigenvectors of your matrix.   Note that $P$ is not orthogonal, although the columns are pairwise orthogonal.
$$    
P =
 \left(  \begin{array}{rrrrrrr}
  1  &  -1  &  -1  &  -1  &  -1  &  -1  &  -1    \\
  1  &  1  &  -1  &  -1  &  -1  &  -1  &  -1     \\
  1  &  0  &  2  &  -1  &  -1  &  -1  &  -1    \\
  1  &  0  &  0  &  3  &  -1  &  -1  &  -1     \\
  1  &  0  &  0  &  0  &  4  &  -1  &  -1     \\
  1  &  0  &  0  &  0  &  0  &  5  &  -1     \\
  1  &  0  &  0  &  0  &  0  &  0  &  6    
\end{array}
  \right).
  $$
The columns of $P$ are of varying lengths;  lengths $ \sqrt{7}, \sqrt{2}, \sqrt{6}, \sqrt{12},..$ All that is necessary to make an orthogonal matrix $Q$ out of this is to divide each column by its length. 
