I got the following problem:
Let $S:\mathbb{M^R}_{3 \times 3} \to \mathbb{M^R}_{3 \times 3}$ be a linear transformation defined by $S(A) = (3A+A^T)/2$ for every matrix $A \in \mathbb{M^R}_{3 \times 3}$
(A): is $S$ an unitary linear operator?
(B): find the minimal and characteristic polynomial for $S$
(C): find the jordan form for $S$
I showed that $S$ is not an unitary operator since $||S(E_{12})|| \neq ||E_{12}||$ but how can I find the characteristic and minimal polynomial for $S$ without calculating a 9 by 9 matrix determinant?
To avoid computing the polynomial, you can just search for eigenvalues and generalized eigenspaces.
$\lambda A = (3A+A^T)/2 \iff 2\lambda A - 3A=A^T\iff (2\lambda -3)A=A^T$
For $\lambda = 2$, you get $A=A^T$ so that's all the symmetric matrices. So the eigenspace has dimension $6$.
For $\lambda=1$, you get $A=-A^T$ so you get all the antisymmetric matrices. So the eigenspace has dimension $3$.
And since $6+3=9$, you've found all eigenvalues and have a basis of eigenvectors.
You should be able to answer the questions now.