Linear around a point in nonlinear differential equation system

[Jingze Wang]

Do we need to calculate the eigenvalues and eigenvectors when the questions just ask us to linearize the system at the point? Or we just find J matrix at that point is sufficient and no need for eigenvectors?

[Chutong(Peng) Judy]

I think at least we need to find the eigenvalues of each critical point.
And we need the corresponding eigenvectors to draw the graph, even though it takes long time to calculate during test.

[Victor Ivrii]

Usually you are not asked just to linearize but to make some conclusions.

If you are asked to classify the point you need to calculate eigenvalues (or at least to say, what is the case) and if we have a repeated eigenvalue to say if there is only one linearly independent eigenvalue.

If you are asked to draw the phase portrait near the point, then finding eigenvectors (for real eigenvalue) is also needed.