(1) Let $A$ be an $n\times n$ real symmetric matrix.
Then, $P^{-1}AP$ is a diagonal matrix for some orthogonal matrix $P$.
Let $L_A:\mathbb{R}^n\ni x\mapsto Ax\in\mathbb{R}^n$.
In this case, we can visualize the action of $L_A$ very easily.
(2) Let $A$ be an $n\times n$ real matrix which has $n$ linearly independent eigenvectors.
Then, $P^{-1}AP$ is a diagonal matrix for some invertible matrix $P$.
Let $L_A:\mathbb{R}^n\ni x\mapsto Ax\in\mathbb{R}^n$.
In this case, we can visualize the action of $L_A$ very easily.
Let $A=\begin{pmatrix}\cos\theta &-\sin\theta\\\sin\theta&\cos\theta \end{pmatrix}$.
Then $A$ is not a $2\times 2$ symmetric matrix if $\theta\neq 0$ or $\theta\neq\pi$.
Then $A$ has no eigenvectors if $\theta\neq 0$ or $\theta\neq\pi$.
But we can visualize the action of $L_A$ very easily.
Let $A$ be an $n\times n$ real matrix which is not symmetric and which doesn't have $n$ lienarly independent eigenvectors.
Can we visualize the action of $L_A$ easily?
Can we decompose $A$ into matrices $A_1,A_2,\dots,A_m$ for which we can visualize the action of $L_{A_i}$ easily?
I have found the answer I was looking for.
https://youtu.be/qTIwqnweaf8