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,

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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?