Suppose I have a matrix $X$ of $n$ assets for $t$ time-periods, where the matrix $X$ is centered.
Let the covariance matrix of $X$ given by $\Sigma =\frac{X X'}{t}$ .
Now suppose I have done a SVD decomposition of $\Sigma$ and $v$ is a matrix whose columns are the eigenvectors. Also suppose $v_1$ is the first eigenvector .
My question is for any individual eigenvector , say for $v_1$, are the elements of that eigenvectors (e.g. $v_{1k}$ for $k$ in $1...n$) equivalent of beta of original assets to the (first) principal component, OR, are the elements of that eigenvector the equivalent of beta of the (first) principal component to the original assets?
If neither is correct, what would be the beta of the original assets to the first eigenvector (i.e. $x_j = a_j + b_j p_1$ where $p_1$ would the $t$ period values corresponding to $v_1$ etc)