TITLE: real eigenvalue
QUESTION [5 upvotes]: Let matrix $A$ be
  $$\begin{bmatrix}
 -5& 1& 0& 0\\
  a &2& 1 &0\\
  0& 1 &1 &1\\
  0 &0&1& 0
\end{bmatrix}$$ 
  where $a$ is a constant between 1 and 3.
Show that the dominant eigenvalue is real.

Thanks a lot!!

REPLY [5 votes]: $A$ is a real tridiagonal matrix. One property of real tridiagonal matrices is this: if the signs of the entries in the upper and lower diagonals are symmetric (i.e. the $(i,\,i+1)$-th and $(i+1,\,i)$-th entries have the same sign for every $i$), then the matrix is similar to a real symmetric matrix and hence all of its eigenvalues are real. Now this is your case here.