? LA and ODE

From: Cheng Cosine (acosine_at_ms13.url.com.tw)
Date: 01/31/05 

Date: Mon, 31 Jan 2005 00:49:37 -0700

Hi:

 Given A*x = b where A is N-by-N matrix, one can analyze this

linear system by spectral analysis. For example, the solution space

is N-dim so once one can find N indep vectors, the solution can be

uniquely expressed. More over, one can use the Fredholm alternative

thm to see if soln exist and is unique or not.

  Now, given dx/dt = A*x+f(t,x), where A is N-by-N matrix.

This is again a N-dim problem. In order to apply the idea from LA,

one tries to write it as: (d/dt-A)x = f(t,x)

But then the "system matrix" is supposed to be (d/dt-A) and the eigen

analysis should be performed to analyze (d/dt-A) to determine the soln

existence and uniqueness. But ODE books analyze only the eigen behaviors

of matrix A. Why is that? To be more explicit,

for A*x = b ==> b vector must be able to be expressed as e-vectors of A

 to have soln.

So for (d/dt-A)x = f(t,x) one should expect something similar. But how?

  Thanks,
by Cheng Cosine
    Jan/31/2k5 UT

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