Construction of Dual Complex Conjugate Biorthonormal Sets

Appendix A. Construction of Dual Complex Conjugate Biorthonormal Sets 

Let us start from a linearly independent set = Ot, t 2. fim) consisting of complex functions the set has the additional property that the overlap matrix A = 0 fl ) is nonsingular, i.e., that A 0. Let y be the similarity transformation, which brings A to classical canonical form k with the eigenvalues on the diagonal and Os and Is on the line above the diagonal  

The eigenvalue kk is nonvanishing, since the matrix A was assumed to be nonsingular. We will further let the symbol kk112 denote a specific square root, which is positive if the number kk is real and positive, and which is situated in the upper half of the complex plane if kk is complex, so that I k l12 0. Even other conventions are, of course, possible. Writing Eq. (A.3) in the form 

Making this construction for all the Jordan blocks occurring in the matrix A, one obtains a specific square root A 1/2, which, substituted in the relation given by Eq. (A.2), leads to the matrix t = A 1/2. We will now show that if the overlap matrix A is symmetric, so that A = A, then the matrix t = A 1/2 has the same property  

It should be observed that the classical canonical form A is by no means symmetric, except in the special case when it happens to be diagonal. The transposed matrix A has the same elements as A on the diagonal, but the Os and Is are now on the line one step below the diagonal. For a specific Jordan block of order p, one obtains from Eq. (A.3) 

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