Suppose we have bases on an inner product space $V$: $\beta = \{ v_1, v_2 \}$ and $\beta' = \{v_1, v_1 + v_2 \}$ such that $\|v_1\| = 1$, $\|v_1 + v_2\|=1$, and $\langle v_1, v_1 + v_2 \rangle = 0$. Suppose that $T: V \to V$ is a linear operator and that the matrix of $T$ in $\beta$ is:
$$[T]_{\beta} = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}.$$
Then, the matrix of $T$ in $\beta'$ is:
$$[T]_{\beta'} = \begin{bmatrix} 2 & 1 \\ 0 & 1 \end{bmatrix}.$$
But these matrices are not transformable into one another by a linear change of coordinates! So do the theorems about Jordan Normal Forms/Diagonalizabilty not hold for matrices of linear operators/tensors?