What is Principal component analysis, minimum-variance portfolio?

Principal component analysis diagonalises a covariance (or correlation) matrix, $S = U\Lambda U^\top$: the eigenvectors are uncorrelated portfolios and the eigenvalues their variances, in decreasing order. The minimum-variance portfolio for a covariance $\Sigma$ is $w = \Sigma^{-1}\mathbf 1/(\mathbf 1^\top\Sigma^{-1}\mathbf 1)$, the fully invested portfolio of smallest variance, $1/(\mathbf 1^\top\Sigma^{-1}\mathbf 1)$.