Diagonalising matrices over different fields examples Let say
$M=\left[ {\begin{array}{cc}
   1 & 1 \\
   1 & 0 \\
  \end{array} } \right]$ and so its characteristic polynomial is $x^2-x-1$, which will be diagonalisable if the field chosen is $\mathbb{R}$ but not if the field is $\mathbb{Q}$.
How can I determine if such matrix is diagonalisable over finite field $\mathbb{F_p}$, for some $p$ prime?
Also if a matrix is diagonalisable over $\mathbb{C}$ but not $\mathbb{R}$ then am I right in thinking that it cannot be diagonalisable over $\mathbb{F_P}$?

You're right, if the matrix is diagonalizable over $\mathbb{C}$ or $\mathbb{R}$, this must not hold for a finite field. Look at your example. I reduce it here to the case $p = 2$. The elements of $\mathbb{F}_2$ are $0$ and $1$. But since
$$ 0^2 - 0 - 1 = 1 = 1^2 - 1 -1,$$
we see that the matrix has no eigenvalues in $\mathbb{F}_2$ and is therefore not diagonalizable over $\mathbb{F}_2$.
In general, also in finite fields, you always have to calculate the characteristic polynomial and investigate the eigenvalues and their algebraic and geometric multiplicities.
