Consider this matrix ( I don't know what mistake i am making in mathjax, I have thought a lot) ( Its a 3 * 3 matrix with 1 st row 2 ,2, 1 )
$$\begin{pmatrix} 2 & 2 & 1 \\ 0 & 2 & -1 \\ 0 & 0 & 3 \\ \end{pmatrix}$$
Question was asked in my algebra quiz that is this diagonalizable over $\mathbb{Q}$?
Eigenvalues of A are 2,2,3 $\in \mathbb{Q}$ . So, why it is not diagonizable.
I seriosly have no idea!
Can you please tell why it is not diagonizable.
$A$ is not diagonalizable over $\Bbb Q$ because it is not diagonalizable at all (i.e. not diagonalizable over $\Bbb C$). It suffices to note that even though $2$ is an eigenvalue with algebraic multiplicity $2$, the geometric multiplicity (dimension of the eigenspace) is $$ \dim \ker (A - 2 I) = 1. $$