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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
May
13
accepted Poincaré lemma and conservative vector fields
May
13
comment Poincaré lemma and conservative vector fields
Of course! So silly of me, thank you. So should I thought as if $X$ satisfies $\frac{\partial X_i}{\partial x_j}=\frac{\partial X_j}{\partial x_i}$, then it is conservative?