Maximum variance = minimum reconstruction error

For any orthonormal W:

‖A − AWWᵀ‖²_F = ‖A‖²_F − ‖AW‖²_F

So keeping the most variance and making the smallest squared reconstruction error select the same subspace.

Covariance and correlation are choices of input geometry. Eigendecomposition and SVD are ways to compute principal axes. They are not four competing names for identical operations.

With repeated eigenvalues, signs and bases can differ — compare subspace projectors, not vectors.