The correlation matrix is defined as =(1/q) ', with column _i of ∈ℝ^(m×q) defined in terms of column _i of and the average as _i=_i-, for i=1,,q.Find the relationship between eigenvalues, eigenvectors of and those of =1/q'.
This is the question I am troubled with. M ends up being a 16384x50 matrix and I really just need to know the relationship between the eigenvalues of C and D. C ends up being a 16384x16384 matrix while D ends up being a 50x50 matrix. I am thinking that maybe their eigenvalues and eigenvectors are the same because the rank of M is 50 meaning that there are up to 50 nonzero singular values, which means there are up to 50 eigenvalues for either C or D. Maybe I was thinking that the eigenvectors of C may be the transpose of the eigenvectors of D.I am not quite sure but would really appreciate some clarification
Both $C$ and $D$ share same set of nonzero eigenvalues.
Let $x$ is an eigenvector of $C$ corresponding to a nonzero eigenvalue $\lambda$. Then we have $$Cx=\frac{1}{q}MM'x=\lambda x.$$ Now multiply both sides (from the left) by $M'$. We get $$D(M'x)=\frac{1}{q}M'M(M'x)=\lambda (M'x).$$ Therefore, $\lambda$ is also an eigenvalue of $D$ afforded by the eigenvector $M'x$.