A and B are two diagonalizable matrices with real coefficients.
1) Prove that if A, B are 2x2 matrices, then they are simultaneously diagonalizable if and only if they have the same eigenspaces or one of the two matrices is multiple to the identity.
2) Prove that this is not true for 3x3 matrices.
I am not able to solve it. Have you any suggestions on how to proceed?
If one of the matrices is a multiple of the identity then we have simultaneous diagonalizability. The same is true if they share the same eigenspaces.
Now let $A$ and $B$ be simultaneously diagonalizable. Then there is a basis $(x_1,x_2)$ of vectors that are eigenvectors for both matrices. If one of the matrices is a multiple of the identity there is nothing to prove. If this is not the case then both matrices have two different eigenvalues with eigenspaces of dimension one. And the eigenspaces coincide.
For the $3\times 3$ case, try to construct to matrices with a basis $(x_1,x_2,x_3)$ of common eigenvectors such that one matrix has two eigenspaces spanned by $x_1,x_2$ and $x_3$, the other has two eigenspaces spanned by $x_1$ and $x_2,x_3$.