5,620 reputation
31028
bio website
location
age
visits member for 2 years, 8 months
seen 49 mins ago

52m
comment Showing unstablity of differential equation.
Since the matrix is $T$-periodic, by Floque theory the fundamental matrix solution is $X(t)=P(t)e^{tB}$, where $P(t)$ is $T$-periodic and $B$ is constant. For eigenvalues of $B$ one has $\lambda_1+\lambda_2=\frac{1}{T}\int_0^T \rm{tr}\, A(t)dt$, from which it follows that the origin is unstable. I am not sure that I can right now answer the second question (shorter than you).
58m
comment Showing unstablity of differential equation.
+1 The eigenvalues of $(A(t)+A(t)^\top)$ work. Unfortunately, this is a very conservative test and does not work always.
1h
comment Showing unstablity of differential equation.
I am really surprised that you continue to give me this kind of arguments. Your answer can be fixed, if you forget about the eigenvalues and work directly with the trace. However, as I already mentioned several times, no conclusion can be made based on the eigenvalues of time dependent matrix. Moreover, as the answer stands right now, it is simply wrong. -1.
2h
comment Showing unstablity of differential equation.
You can find an example in Wu, 1974, A Note on Stability of Linear Time-Varying System.
2h
comment Showing unstablity of differential equation.
I am not aware of any. However, it does not prove anything, since, as I mentioned, you cannot invoke these kind of arguments dealing with time-dependent matrices.
2h
comment Showing unstablity of differential equation.
No, this is just handwaving. There are examples of time dependent matrices that have constant negative eigenvalues. And the oring is unstable.
3h
comment Showing unstablity of differential equation.
This is incorrect. You cannot invoke the eigenvalues while studying problems with non-constant matrices.
Dec
14
answered Good Pre-Calculus book?
Dec
14
comment Good Pre-Calculus book?
Actually, the name is Shen
Dec
12
comment Genius mathematicians who never published anything
Perelman published quite a few papers in a usual way. Even before the whole story about Poincare's conjecture he was offered a number of positions in different US universities.
Dec
10
reviewed Approve How do you reverse $\frac{100n(n+1)}{2}=c$ to find n given c?
Dec
10
revised Limit of solution of linear system of ODEs as $t\to \infty$
added 484 characters in body
Dec
10
comment Limit of solution of linear system of ODEs as $t\to \infty$
@User267845467 For the second question: yes. For the first one: it definitely requires some proof. I will try to outline it.
Dec
10
answered Limit of solution of linear system of ODEs as $t\to \infty$
Dec
9
awarded  Popular Question
Dec
9
reviewed Approve prove that if n is odd then 5n +3 is even
Dec
9
reviewed Approve Laplace Transformation using Heaviside functions
Dec
8
awarded  Caucus
Dec
8
reviewed Approve Evaluate the contour integral
Dec
8
reviewed Approve Finding the general solution to X'=AX with A = [0 1 0; 1 0 0; 1 1 1]?