Why do eigenvalues exclusively form the main diagonal in a diagonalizable matrix?

Question

So, why do eigenvalues exclusively form the main diagonal in a diagonalizable matrix?

If we have $n\times n$ matrix ($n$ being a natural number) that is diagonalizable, why is it eigenvalues (exclusively eigenvalues) that make up the main diagonal?

Answer 1

It is not true, I think you mean diagonal matrix instead of diagonalizable. As an example you can take the 2x2 matrix $A$=$\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}$, whose eigenvalues are $3$ and $-1$.

If you meant diagonal then you can check it almost from the definition.

Another way to check it is thinking that a matrix must have the same eigenvalues that its asociated diagonal matrix and then check the characteristic polynomial