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Two optimal orthogonalization processes are devised to orthogonalize, possibly approximately, the columns of a very large and possibly sparse matrix
A ∈ ℂ
n
×
k
. Algorithmically the aim is, at each step, to optimally decrease nonorthogonality of all the columns of A. One process relies on using translated small rank corrections. Another is a polynomial orthogonalization process for performing the Löwdin orthogonalization. The steps rely on using iterative methods combined, preferably, with preconditioning which can have a dramatic effect on how fast nonorthogonality decreases. The speed of orthogonalization depends on how bunched the singular values of
A
are, modulo the number of steps taken. These methods put the steps of the Gram-Schmidt orthogonalization process into perspective regarding their (lack of) optimality. The constructions are entirely operator theoretic and can be extended to infinite dimensional Hilbert spaces.