Tell if $A$ is diagonalized using it's characteristic and minimal polynomials

Question

$$A= \left( {\matrix{ 2 & 1 \cr 1 & 2 \cr } } \right)$$

I already calculated that $f_A(x) = (x-3)(x-1)$. Also, the minimal polynomial must be also $(x-3)(x-1)$.

How can I use this information to determine if $A$ is diagonlized?

Answer 1

Let $v_1$ and $v_2$ two eigenvectors associated to the eigenvalues $3$ and $1$ respectively. $B=(v_1,v_2)$ is a basis for $\Bbb R^2$ and the matrix $A$ is similar to the diagonal matrix $\operatorname{diag}(3,1)$ relative to the basis $B$. A generalization of this is

Theorem If a matirx $A\in M_n(\Bbb R)$ has $n$ distinct eigenvalues then $A$ is diagonalizable.

Answer 2

your matrix $A$ is symmetric. for symmetric matrices, spectral decomposition theorem says that they are diagonalizable whether the eigenvalues are repeated or not.

Answer 3

A matrix or linear map is diagonalizable over some field if and only if its minimal polynomial is a product of distinct linear factors on the same field.