When this matrix is diagonalizable? When this matrix is diagonalizable? ($a_i \in \mathbb{R}$)
$$
\begin{pmatrix}
&&&a_1\\
&&a_2&\\
&\ddots&&\\
a_n&&&\\
\end{pmatrix}
$$
I think I should probably consider characteristic polynomial of this matrix, and if all roots are simple, then the matrix is diagonalizable.
UPD: Also I suppose that if $\forall a_i \neq 0$ then the matrix is diagonalizable, but I can't prove that.
The matrix is anti-diagonal, of course.

Let $e_i$ be the standard unit vectors, $i=1\ldots n$.
The two-dimensional subspaces spanned by $e_i$ and $e_{n+1-i}$, $i = 1 \ldots \lfloor n/2 \rfloor$ and (if $n$ is odd) the one-dimensional subspace spanned by $e_{(n+1)/2}$, are invariant under your matrix, so everything reduces to the two-dimensional case.
If $a_1, a_2 \ne 0$, $ \pmatrix{0 & a_1\cr a_2 & 0\cr}$ has two distinct eigenvalues $\pm \sqrt{a_1 a_2}$, therefore is diagonalizable.  Of course if $a_1 = a_2 = 0$, you have the $0$ matrix which is diagonalizable.  However, if one of $a_1$ is $0$ and the other is not, the matrix is not diagonalizable (the only eigenvalue is $0$, but the null space is one-dimensional).
Back to the general case: the matrix is diagonalizable unless for some $i$, one of $a_i$ and $a_{n+1-i}$ is $0$ and the other is not.
