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 Feb 23 revised On the $\omega$-limit set of a trajectory converging to a submanifold added 366 characters in body; edited tags Feb 22 asked On the $\omega$-limit set of a trajectory converging to a submanifold Jan 16 awarded Yearling Oct 11 comment Question about phase portrait and invariant subspaces @FemaleTank by definition all trajectories are invariant subspaces. Take any point in an orbit, take the action of the flow on that point in forward and backward time, it will give you the entire trajectory. Thus, eigenvectors are not the only invariant subspaces, however, they are important since you can obtain them explicitly without integrating the differential equation. Oct 11 accepted A function that is tangent to both axes Oct 11 comment Question about phase portrait and invariant subspaces First, I think that as you draw the directions, if $x_0\in L_1$ then $\lim_{t\to-\infty}e^{At}x_0=0$, right? (note the $-\infty$). Second, the same conclusions that you have for one eigenspace, say $L_1$ are applicable to $L_2$. So they you prove invariance in one, applies the same for the other just by changing time direction. Perhaps you can give more details if this is not enough clarification. Sep 15 revised A function that is tangent to both axes added 65 characters in body Sep 15 asked A function that is tangent to both axes Sep 14 revised A simple linear algebra question. added 8 characters in body Sep 12 answered A simple linear algebra question. Sep 12 comment A simple linear algebra question. It depends on the number of eigenvectors there are associated to the eigenvalue. It may happened that there are 1,2, or 3 linearly independent eigenvectors, which in turn will generate the eigenspace. For this, Jordan canonical form may be helpful. Aug 12 revised Conditions to preserve Laplacian matrix added 118 characters in body Aug 12 comment Conditions to preserve Laplacian matrix No, those are the principal properties that a Laplacian satisfies, in fact $\ker L=1_n$ is a consequence of $L1_n=0$ plus multiplicity assumptions. The definition of $L$ that I use is $L=D-A$ where $D$ is a diagonal "degree" matrix and $A$ is an adjacency matrix. Sorry, my syntax was misleading. Aug 12 comment Conditions to preserve Laplacian matrix Indeed, even more, I may assume (generically) that $\ker L=\left\{ 1_n\right\}$. However, concluding things from there is where I am stuck. Aug 12 comment Conditions to preserve Laplacian matrix Checking again your answer, I think it is not true since $W$ is or size $r\times n$. Then, the equation $Wx=0$ has non zero solutions. Aug 11 comment Diagonalization of A Dynamical System with Multiple Zero Eigenvalues exactly which part of Perko's book are you referring to? Under no extra assumption, what you claim may not be true (for example, depending on the multiplicity of the eigenvalues). It may interest you a result regarding the topological conjugacy (and decoupling) of systems having a center manifold, see e.g., staff.science.uu.nl/~kouzn101/NLDV/Lect12.pdf Aug 5 revised Finding inhomogeneous solution given homogeneous solution. Correct title Aug 5 suggested approved edit on Finding inhomogeneous solution given homogeneous solution. Aug 4 revised Conditions to preserve Laplacian matrix added 2 characters in body Aug 4 asked Conditions to preserve Laplacian matrix