Describe and sketch the vector field $Ax$. $A$ is symmetric and skew-symmetric.

Question

$A$ is a $2\times2$ matrix.

1. $A$ is a symmetric matrix with the eigenvalues $\lambda_1$ and $\lambda_2$ (consider cases of all sign combinations).
2. $A$ is skew–symmetric.

I think the vector field of symmetric matrix is an irrotational field and converges to the eigenvector if the eigenvalue is negative. The vector field of skew-symmetric matrix is a divergence-free field.

Answer 1

Hint: Note that the flows in this vector field are solutions to the system of linear differential equations $\frac{dx}{dt} = Ax$. It suffices then to consider the eigenvalues of the matrix $e^{At}$ (where $t > 0$).

Note that symmetric matrices have purely real eigenvalues, and skew-symmetric matrices have purely imaginary eigenvalues.