TITLE: Can I quickly determine the eigenvalues of this matrix?
QUESTION [5 upvotes]: I am working on observability and detectability in controls and I ran across this example that I didn't understand. The author deliberately sought the form of this matrix, because of its "block-form" in order to quickly find the eigenvalues
\begin{bmatrix}
    l_{11}  & -1  & -1 & 0 & 0 \\
    0 & -1  & 0 & 0 & 0 \\
    0 & -1  & -1 & 0 & 0 \\
    0 & -.1 & -2 & l_{42} & -.1 \\
    0 &  1  &  2 & 0 & -.2
\end{bmatrix}
The author was then able to state the eigenvalues were $\{l_{11}, -1, l_{42}, -.2\}$ 
I was under the impression that I could only determine the eigenvalues via  a matrix diagonal if the matrix was upper/lower triangular?

REPLY [9 votes]: This matrix is block lower triangular:
$$\left[\begin{array}{ccc|cc}
    l_{11}  & -1  & -1 & 0 & 0 \\
    0 & -1  & 0 & 0 & 0 \\
    0 & -1  & -1 & 0 & 0 \\
\hline
    0 & -.1 & -2 & l_{42} & -.1 \\
    0 &  1  &  2 & 0 & -.2
\end{array}\right]$$
From the bottom-right block we see, that eigenvalues are $l_{42}$ and $-0.2$. 
The upper-left block is again block upper triangular
$$\left[\begin{array}{c|cc}
    l_{11}  & -1  & -1\\
\hline
    0 & -1  & 0\\
    0 & -1  & -1 \\
\end{array}\right]$$
and has eigenvalue $l_{11}$ and double eigenvalue $-1$.