If I have
x' = Ax
where A is the matrix below
then the direction field of the solutions would have everything moving towards (0, 0) as t --> infinity. My question is: how would I describe the behavior as t --> infinity in words? Is it just "stable"?
A much more important question is: if I have a 2X2 matrix A, and the eigenvalues of that A are both greater than zero, then what would the direction field look like?
Again, an even MORE important question is: if I have A as a 3X3 matrix, and so have three eigenvalues for it, and two of them are the same.... writing the general solution wouldn't be just
x = (constant1)(eigenvector1)(e^eigenvalue1t) + (constant2)(eigenvector2)(e^eigenvalue2t) + (constant3)(eigenvector3)(e^eigenvalue3t)
would it? That just doesn't seem right, and I can't get my answer doing it that way to match the book's answer.
While I'm at it, let's put in another question here. I'm sure it's a simple question, but my class is at eight in the morning, and lasts forever, so... I drift.
Anyway, how could I write an eigenvector in terms of its eigenvalue? What I'm trying to do is show that, considering the 2X2 system x' = Ax,
(A - r1I)E1 = (r1 - r2)E1
where r1 and r2 are the eigenvalues and E1 is the eigenvector corresponding to r1.