If a matrix $\bf X = UV'$ such that $\bf UU' = \Sigma$, $\bf{VV'} = I$, $\bf V\Sigma = X'XV$ and $\bf U = XV$ where $\bf \Sigma$ is a diagonal matrix.
From the equality $\bf V\Sigma = X'XV$, we can deduce that columns of $\bf V$ is the eigenvectors of the matrix $\bf X'X$. Similarly, we can deduce that $\bf \Sigma U = XX'U$, which makes $\bf U$ the eigenvectors of $\bf XX'$. But I am told that $\bf U$ is the eigenvectors of $\bf XX'$ scaled by the eigen values of $\bf XX'$. Where I am wrong ?