The ability to readily visualize a matrix-vector multiplication is of great value in physics. Here is a little problem to test how comfortable you are with matrix-vector multiplication and using that to find eigenvectors and their eigenvalues. Give it a try. (Don't look at anything after the break unless you are stuck.)
Using just guessing and matrix-vector multiplication, find one or more eigenvectors of this matrix:
$$ \begin{matrix} 1 & 1 &1 &1 \\ 1 & 1 & 1 &1 \\ 1 & 1 & 1 &1 \\ 1 & 1 & 1 &1 \\ \end{matrix} $$
As you may know, the sum of the eigenvalues is equal to the trace of the matrix. Is there one eigenvector that "exhausts the trace"?
One of the eigenvectors I am hoping you found is (1,1,1,1). Its eigenvalue of 4 is equal to the trace so the other 3 have to add up to zero. Maybe they are all zero?
How about:
ReplyDelete(1,0,-1,0), (0,1,0,-1) and (-1,1,-1,1)?
I am not really sure about this so they might be all wrong, let me know!
Yes! I think that those are mutually orthogonal, and they all have eigenvalues of zero. There should be one more the eigenvalue of which is equal to the trace of the matrix.
DeleteWait... unless I'm doing it wrong, those eigenvalues are zero... what does that mean about the matrix?
DeleteSum of the eigvalues is equal to the trace...
Delete(What matrix are you asking about?