Consider the following square matrix:
$A= \left( \begin{matrix} \lambda_1 & a_{1,2} & & ... & a_{1,n} \\ 0 & \lambda_2 & & ... & a_{2,n} \\ . & & . & & . \\ . & & & . & a_{n.1,n}\\ 0 & ... & & & \lambda_n\\ \end{matrix} \right) $
Characterize, in terms of the coefficients $a_{ij}$ if it is similar to a diagonal matrix when:
i) All the $\lambda_i$ are equal to $\lambda$.
ii) All the $\lambda_i$ are different from each other.
iii) $\lambda_1 \not= \lambda_2= \lambda_3=...=\lambda_n$.
I think I've solved it but I'm not completely sure...so please tell me if I'm not. Thank you for your kind help!
i) We need to have $dim(ker(\lambda I - A))=n$. Since it's a $n \times n$ matrix it must be $a_{ij}=0 \ \forall i,j=1,...,n$ so it is already a diagonal matrix.
ii) Already seen in this website (answer is: always).
iii) This means $dim(ker(\lambda I - A))=n-1$, so there must be an $a_{ij}\not=0$ and we then there might be other non-zero coefficients either on the $i^{th}$ row or on the $j^{th}$ column (not both or the dimension of the kernel would decrease.