I am trying to show that this matrix:
$$L=\begin{bmatrix} f_0 & f_1& \cdot & \cdot & f_n\\ p_0 & 0 & \cdot & \cdot & 0\\ 0 & p_1& \cdot & \cdot & 0\\ \cdot & \cdot& \cdot & \cdot & \cdot\\ 0 & 0& \cdot & p_{n-1} & 0 \end{bmatrix}\mbox{ , } 0\leq f_i \mbox{ and }0<p_i<1.$$
commonly known as the Leslie matrix is diagonalizable.
I know a $n \times n$ matrix is diagonalizable if there exist $n$ linearly independent eigenvectors. I also know two different eigenvalues have linearly independent eigenvectors. So my approach was to show the characteristic equation of the Leslie matrix has $n$ different solutions, but here I got stuck. Can somebody help me?
Here is a simple counterexample with a $3 \times 3$ matrix:
$$L=\begin{bmatrix} 1 & 5& 3\\ 1 & 0& 0\\ 0 & 1& 0 \end{bmatrix}$$
with eigenvalues $-1$ (order of multiplicity $2$) and $3$.
The eigenspace associated with eigenvalue $-1$ is obtained as solution of
$$\begin{cases}2x&+&5y&+&3z&=&0\\x&+&y&&&=&0\\&&y&+&z&=&0\end{cases}$$
which gives $\begin{bmatrix}x\\y\\z\end{bmatrix} \ = \ k \ \begin{bmatrix}\ \ 1\\-1\\ \ \ 1\end{bmatrix}.$
Thus, this eigenspace is only one dimensional. As this dimension is less than the order of multiplicity of the corresponding eigenvalue, matrix $L$ is not diagonalizable.
Remark 1: $kL$, with any positive real $k$, is a counterexample as well.
Remark 2: matrix $L$ is a "companion" matrix : see the Wikipedia article (http://www.math.wsu.edu/faculty/watkins/seminar/pdfiles/fiedler_companion.pdf) where it is explicitly said: "In general, the companion matrix may be non-diagonalizable.". But the difficulty is to find such a matrix with positive coefficients.
Remark 3: a companion matrix can have different equivalent forms: one can find the (opposite of the) list of polynomial coefficients on
the rightmost column (as in the Wikipedia article)
upper row (in direct correspondence with Leslie matrices, as the convention taken in Matlab (https://fr.mathworks.com/help/matlab/ref/compan.html)),
bottom row.
Remark 4: the counterexample given by @quasi is, up to a multiplication by 2 (which does not affect the "non-diagonalizability" property), a "companion matrix".
If you are interested by companion matrices, here is a very interesting discussion by Cleve Moler, the founder of Matlab: (http://blogs.mathworks.com/cleve/2013/12/23/fiedler-companion-matrix/)