Conditions for diagonalizability of $n\times n$ anti-diagonal matrices Let $A$ be an $n\times n$ anti-diagonal matrix: $a_{i,j}=0$ unless $i+j=n+1$. 
A) When is $A$ diagonalizable (what are the conditions on the $a_{i,n+1−i}$)?
B) Find the eigenvalues and eigenvectors of $A$ for $n=7$ and $a_{i,n+1−i}=i^2$.
Not sure how to get started on part (A) - I've played around with $\det[A-\lambda I]$ for $A$, an $n\times n$ anti-diagonal and didn't come up with any specific conditions for diagonalizability.
I know we need to produce a basis of eigenvectors of $A$ -- showing that $A$ is similar to a diagonal matrix $D$.
For part (B), all the anti-diagonal entries take the value $i^2 = -1$.  I'm currently trying to compute this by brute force - and got a bunch of -$\lambda$'s on the main diagonal, $-1-\lambda$ in the middle entry of the matrix, and then $\det[A-\lambda I]$ is something like $p(\lambda)$ = $-\lambda^7 - \lambda^{6}$.  Solving for the eigenvalues (and then computing the corresponding eigenvectors) doesn't really follow at this point, so I'm guessing there is a trick that I need to use.
Thanks in advance,
 A: if you make $A$ symmetric, then $A$ would certainly be diagonalizable. that is you make $a_{1n} = a_{n1}, a_{2n-1} = a_{n-12}, \cdots.$  it is enough for matrix $A$ to be normal, i.e., $AA^* = A^*A,$ where $A^*$ is the complex conjugate of $A^T,$ for it to be diagonalizable.  if you look at a $3 \times 3$ matrix $A = diag(a, b, c)\ $, you see that 
$AA^* = diag(|a|^2, |b|^2, |c|^2), A^TA = diag(|c|^2, |b|^2, |a|^2)$  now you only need that $|a|  = \pm |c|$  for $A$ to be diagonalizable over $C$ which is less stringent than $a = c$ 
see if you can verify that  $|a_{1n}| = \pm |a_{n1}|, |a_{2n-1}| = \pm |a_{n-12}|, \cdots$ is necessary and sufficient for $A$ to be diagonlizable.
edit:
hint for part (b). i will use $e_1 = (1,0,0,0,0,0,0)^T,\cdots$ for the standard basis vectors in $R^7.$
here we have $n = 7$ and $Ae_1 = 49e_7, A e_2 = 36e_6,\cdots, Ae_6 = 4e_2, Ae_7 = e_1$ observe the pairing $(Ae_1, Ae_7), (Ae_2, Ae_6), \cdots$ and a singleton $Ae_4 = 16 e_4$
 last one already tells you that $4$ is an eigenvalue and a corresponding eigenvectors is $e_4.$ the rest make up three $2 \times 2$ blocks. 
can you find the eigenvalues and the corresponding eigenvectors for each of the three pairs?
