(BCS) pairing theory , and its generalization by
are based on fermion wave functions that have the form of
fermion-pair condensates, i.e.,
The Bloch-Messiah-Zumino theorem [3,4] provides the
link between the two forms of state ,
Eqs. (1) and (3), by stating that every complex
antisymmetric matrix can be brought by a unitary transformation into its
canonical form, i.e.,
When using states (1) in applications beyond the mean-field
approximation, and in particular in the generator coordinate method (GCM)
[5,6,7,8], the matrix elements and
overlaps depend on the product matrix C+C'. For example, the
overlap of two states (1) reads [9,6,8]
In the present paper, I generalize results of Ref.  by deriving canonical forms of two arbitrary complex matrices C and C' in a common canonical basis. These results are not restricted by any conditions on matrices C and C'.
Let us begin by recalling the notion of the Jordan form
(see e.g. ) of an arbitrary complex matrix.
Focussing our attention on the matrix C+C', the vectors
defining its Jordan basis can be arranged in columns
of matrix W, and one has
Basis vectors belonging to a given block I form the so-called
Jordan series of length L. The series starts with the basis vector
called the series head, and ends with an eigenvector of C+C'. The
whole series is uniquely determined by the series head, because the
remaining basis vectors in the series can be obtained by a repeated
action of C+C' on the series head.
The basis vectors in a given series are not unique, because a linear
combination of these vectors may give another valid series head, and
leads to the same Jordan canonical form. Explicitly, this transformation
According to the Jordan construction, the whole space in which acts matrix C+C' splits into subspaces spanned by the Jordan series. The number of eigenvectors of C+C' equals to the number of different series, or to the number of Jordan blocks, and is in general smaller than the dimension of the matrix C+C'. Some matrices (hermitian or not) can be fully diagonalized, i.e., they have numbers of eigenvectors equal to their dimensions. This corresponds to the case when all the Jordan series have the length equal 1.
One calls two blocks degenerate, or two series degenerate, if they have the same diagonal number DI, and they have the same length L. The latter condition is very important, because only degenerate series defined in such a way can be mixed; this is an analogue of the possibility to mix degenerate eigenvectors of a matrix which can be fully diagonalized. If two series have different lengths then vectors of a longer series cannot be admixed to those of the shorter series, even if the series have the same diagonal number DI. If the matrix can be fully diagonalized, then all the series have length 1, the number of eigenvectors equals to the dimension of C+C', and transformation (12) reduces to the possibility of arbitrarily normalizing every eigenvector.
After these necessary preliminaries, let us proceed with presenting
the main results of the present paper.
Multiplying from the left-hand and right-hand sides the
eigen-equation (9) by W-1, and then transposing, we
One can see now that matrix C+C' has another equivalent set of Jordan
We can now analyze cases of different degeneracies of the Jordan blocks. The arguments given below closely follow proofs presented in Ref. , only with the degeneracies of eigenvalues replaced by the degeneracies of the Jordan blocks.
Let us first suppose that C+C' has a non-degenerate Jordan
block. Then, in this block the basis vectors V must be connected
with the basis vectors W by transformation (12), i.e.,
One can set this subspace aside, and assume from now on that C+C'is non-singular and has an even dimension. In this case, matrix C+C' cannot have any non-degenerate Jordan block, and hence Jordan blocks must appear in degenerate pairs. (In odd dimensions, C+C'must have at least one null eigenvalue, which can be separated, and the remaining matrix can be treated in the even dimension).
In the present considerations, it is enough to consider only pairs of degenerate blocks; had the higher degeneracies of the Jordan blocks occurred, one could have considered one pair after another, and at each step one could reduce the dimension of the problem. This is possible here, and has not been possible when considering degenerate eigenvalues in Ref. , because the whole space can be separated into the subspaces corresponding to the Jordan blocks, while it cannot be separated into subpaces corresponding to the eigenvalues.
Let us now consider a pair of degenerate Jordan blocks, each block having length L and the common diagonal element DI= . One can adopt here the standard notation that originally pertains to the canonical pairs, namely, we denote the indices of the two degenerate blocks by I and . Similarly, indices inside these two blocks are denoted by k=1, 2,...,L and =1, 2,...,L, respectively. Note that vectors in these two blocks form series, i.e., they are arranged in a specific order; therefore a vector at a given position must be associated with the vector at the same position in the second block.
Since for matrix C+C' two equivalent Jordan bases exist, W and V,
vectors in series V must be linear combinations of those in
In the pair of degenerate Jordan blocks,
this leads to the following relations between the two series:
Therefore, the canonical form of the C+ matrix reads
Since the canonical basis of C+ is the same as the Jordan
basis of C+C', matrix C' must in the very same basis assume
an analogous canonical form:
Equations (22) and (24) complete the proof of the canonical forms of two arbitrary complex antisymmetric matrices C and C'. Both these matrices can be simultaneously transformed by matrix W (in general non-unitary) into the block-diagonal forms with non-zero elements only between pairs of degenerate Jordan blocks.
Needless to say, whenever matrix C+C' can be fully diagonalized,
which was the case in Ref. , both matrices C and C'acquire in the canonical basis the standard canonical forms
analogous to Eq. (6), i.e.,
In Ref.  it was noticed that an incorrect conjecture was formulated in Ref. , namely, the conjecture that the simple forms of Eqs. (27) and (28) can always be achieved. In the present study we have seen that these simple forms occur only when matrix C+C' can be fully diagonalized. In fact, this is the case which occurs most often in applications. Therefore, let us now discuss conditions for the full diagonalization of C+C'.
In the applications given in Ref. , the full
diagonalization of matrix C+C' was secured by using a model in
which matrices C were time-even,
In the general presentation of the present paper, conditions (29) and (30) can be formulated as follows: If there exist a unitary antisymmetric matrix UT such that Eq. (29) holds for C and C', and at the same time at least or is positive-definite, then matrix C+C' can be fully diagonalized, and the simple canonical forms (27) and (28) exist. The proof of this statement has been given in Ref.  (Appendix C), and will not be repeated here.
The positive definiteness of
is a required condition,
because the hermitian square-root of
Unfortunately, this condition cannot be released, i.e., if both
are not positive definite, it may happen
that matrix C+C' cannot be fully diagonalized.
An example of such a situation is provided by the following
two 44 matrices:
However, for any small but non-zero imaginary part of a, matrix C+C' can be fully diagonalized. Therefore, this example also shows that the positive definiteness of (time-even) matrices or is only a sufficient condition for the full diagonalization of C+C'= , but it is not necessary. Moreover, it is clear that matrix C+C' cannot be diagonalized for a= , because in the limit of 0 two eigenvectors of C+C' become parallel. This illustrates the difficulty of diagonalizing C+C'numerically for small values of ; the task is then bound to become ill-conditioned.
In the GCM, matrices C are most often obtained from solutions of the Hartree-Fock-Bogoliubov (HFB) or Hartree-Fock+BCS  equations for time-even states. In these cases, matrices are diagonal in the HFB or BCS canonical bases  (composed of pairs of time-reversed states), and their eigenvalues are equal to vm/um=cm, where vm and um are the standard quasiparticle amplitudes of Eq. (5). Here, the canonical pairs are defined by the time reversal, and therefore the eigenvalues cm, can have, in principle, arbitrary signs.
However, in the BCS method (with a constant gap parameter ) all these quasiparticle amplitudes are positive, and hence all the resulting matrices are positive definite, thus fulfilling the sufficient condition for the full diagonalization of C+C'= . In fact, quasiparticle amplitudes of different signs rarely occur in nuclear physics applications, cf. Ref. . This is so, because typical pairing forces couple the time-reversed states, and, in general, are always attractive. This shows that the Jordan structures discussed here cannot be expected to be frequently encountered, and most often one will deal with the standard canonical forms of Eqs. (27) and (28), in which the only non-zero matrix elements are adjacent to the main diagonal.
In summary, I have shown how to extend the results of Ref.  in order to construct canonical basis in which two arbitrary complex antisymmetric matrices C and C' acquire simultaneously canonical forms. This construction completes the generalization of the classic Bloch-Messiah-Zumino theorem to the case of non-diagonal matrix elements calculated between fermion-pair condensates.
The critical reading of the manuscript by S.G. Rohozinski is gratefully appreciated. This research was supported by the Polish Committee for Scientific Research, Contract No. 2 P03B 040 14.