I found a surprising property of the eigenvectors of the matrix $A = X^T X$ and $B = XX^T$ experimentally. Let $X$ be $n \times d$ with $n > d$. Then $A$ and $B$ are psd matrices. The eigenvalues of $A$ and $B$ are equal (except $B$ has additional zero eigenvalues). Take an eigenvalue $\lambda$ of $A$ and $B$, and take the corresponding eigenvector $v\_A$ ($d \times 1$) and $v\_B$ ($n \times 1$) of $A$ and $B$. I found that $|X v\_A |\propto |v\_B|$, where the absolute value is taken of each element of the vector. Is this result well known or can anybody confirm / prove this?