I'm trying to find a matrix N similar to the scalar matrix M = $ \begin{pmatrix} a & 0 \\ 0 & a \\ \end{pmatrix} $ Such that $M = ANA^{-1}$. I have no idea where to start? To me it doesn't appear such a matrix would exist, but I don't know how to prove it.
Also, if I were to change the matrix M to $ \begin{pmatrix} a_1 & 0 \\ 0 & a_2 \\ \end{pmatrix} $ would it then be possible to find N similar to M?
It means that you want :
$M = a*I =N*A*N^{-1} => A = a*N^{-1}*I*N = a*I = M$ So in this case there is only one solution.
For the second case, all you have to do is find a diagonalizable matrix with the coefficient of M as eigenvalues :
$X_A(X)=X^2 -tr(A)*X +det(A)=(X-a_1)*(X-a_2)$ $ =X^2 -(a_1+a_2)*X +a_1*a_2$ For instance, we can take: $$N = \begin{pmatrix} a_1 +a_2 & -a_2 \\ a_1 & 0 \\ \end{pmatrix}$$
I'll let you find the good A matrix, which exists at least.