I'm using a Householder routine to numerically diagonalize nx2n matrices. The resulting diagonalized matrices are then used in an eigenmode process, which will occasionally just produce "noise". I've noticed that in these cases, the Householder routine is returning matrices which possess some zero values on the diagonal entries, e.g.:
$$R=\begin{bmatrix}a&*&*&*&*&*&*&*&*&*&*&*\\0&b&*&*&*&*&*&*&*&*&*&*\\0&0&c&*&*&*&*&*&*&*&*&*\\0&0&0&d&*&*&*&*&*&*&*&*\\0&0&0&0&0&0&*&*&*&*&*&*\\0&0&0&0&0&0&*&*&*&*&*&*\end{bmatrix}$$
Is there anything inappropriate about this type of this solution resulting from using Householder's routine? If so, does this form (with the zero entries on the diagonal) suggest anything about the original matrix?