Hi, I was wondering if you guys would be able to help me out with a problem I've been struggling with for a few days.
Assuming M = 2x2 matrix with a single eigenvalue c, with eigenvector x, and that y is a vector not equal to 0 which is not an eigenvector. How can I show that x and y are linearly independent?
M being a matrix with a single eigenvalue c, with eigenvector x you have
Mx = cx
Suppose x and y are not linearly independant. There exists a real k such as y=kx
My = M(kx) = k(Mx) = k(cx) = c(kx) = cy
Then y would be an eigenvector of M which is not the case