A new orthogonalization procedure with an extremal property

Various methods of constructing an orthonomal set out of a given set of linearly independent vectors are discussed. Particular attention is paid to the Gram-Schmidt and the Schweinler-Wigner orthogonalization procedures. A new orthogonalization procedure which, like the SchweinlerWigner procedure, is democratic and is endowed with an extremal property is suggested. PACS No: 02.10.Sp, 31.15.+q e-mail:[email protected] email:[email protected] e-mail:[email protected] 1 Constructing an orthonormal set out of a given set of linearly independent Vectors is an age old problem. Among the many possible orthogonalization procedures, the two algorithmic procedures that have been extensively discussed and used in the literature are (a) the familiar Gram-Schmidt procedure [1] and (b) a procedure which is referred to as the Schweinler -Wigner procedure [2], particularly in the wavelet literature [3]. ( This method is known among the chemists as the Löwdin orthogonalization procedure [4]. Mathematicians attribute it to Poincaré. Schweinler and Wigner themselves trace its origin to a work of Landshoff [5]. Eschewing the question of historically correct attribution, we shall continue to refer to it as the Schweinler-Wigner procedure ) An intrinsic difference between the two procedures is that while the Gram-Schmidt procedure, by its very nature, requires one to select the linearly independent vectors sequentially, the Schweinler-Wigner procedures treats all the members of the set of linearly independent vectors democratically. The significance of the work of Schweinler and Wigner lies not in introducing a new orthogonalization method the method was already known but rather in introducing a positive quantity m, to be defined shortly, which discriminates between various orthogonalization procedures. They showed that m is a maximum for the Schweinler-Wigner basis. In this letter we pose and answer the question as to what is the orthogonalization procedure which minimizes m. This new orthogonalization procedure, like the Schweinler-Wigner procedure, also turns out to be completely democratic in that it treats all the linearly independent vectors on the same footing. The quantity m was introduced by Schweinler and Wigner in a some what ad-hoc manner. We reformulate their procedure in a way so as that the quantity m appears in a natural way and can be useful in a wider context than that for which it was introduced. In particular, this reformulation enables us to quantify the notion of an orthonormal basis which brings any Hermitian operator in to a maximally off-diagonal form. Let v1, · · · , vN denote a set of N linearly independent vectors. Let M denote the associated Gram matrix :Mij = (vi, vj). M is a positive definite Hermitian matrix. Define z = vS , (1)