Hey. What can be said about all the normal nxn matrices that have exactly 1 eigenvalue? I'm interested in the case where the entries are in C (complex #'s). what sort of generalizations can we make?
Trambolin, looking at your first matrix the characteristic equation is
[tex](1-\lambda)(\lambda^2-1) = 0 \Rightarrow \lambda = \pm 1[/tex]
That permits two unique eigenvalues, not one.
Besides if you simply choose the coordinate system aligned with the eigenvectors, in that coordinate system the matrix (call it A) will be proportional to the identity matrix. Then the matrix [tex]A-\lambda I[/tex] will vanish in that coordinate system for some complex number [tex]\lambda[/tex], but then it would have to vanish in all coordinate systems and therefore the matrix is proportional to the identity matrix.