This is likely a silly question so sorry in advance. However, I am wondering if I am right in thinking that if zero is an eigenvalue, then some dimension must be lost. My understanding is that eigenvectors are the vectors that are only scaled when some matrix A is applied. Then, eigenvalues are the amount those vectors are scaled. So, therefore is it true that if an eigenvalue is zero, that vector has been sent to the origin and therefore the dimension of the image is smaller than the original dimension?
Would that not also imply that any vector in the kernel of a transformation is an eigenvector?