Given a matrix $X$. Calculating the Eigen values of $XX^T$ and using the ratio of maximum and minimum eigen values normally gives the condition number of the matrix.

If $X$ contains $M$ observations each with $N$ points. How minimizing or maximizing the ratio of eigenvalues lead to mutually independent/orthogonal observations in $X$ over multiple iterations. Considering we add an observation into $X$ if it reduces or maximizes the ratio.

Will the newly added observation in $X$ be independent from previous ones. How can it be ensured. What constraints can ensure such a thing.

Thanks.