I've got a matrix $A = \begin{bmatrix}1&-1&0\\ -1&2&-1\\0&-1&1\end{bmatrix}$ and am asked to determine if it is diagonalisable. I find the eigenvalues to be $3, 1$ and from this the question says we should be able to deduce whether A is diagonalisable or not - without any more work. I know that if
$dim(E_1) + dim(E_3) = 3$
then the matrix is diagonalisable. However, I don't know the dimensions of the eigenspaces without first finding them. I'm not really sure what to do, is the multiplicity of the eigenvalues used? If so what is the multiplicity of them (what does it mean)?
Thanks