Jordan block

When the matrix has non-zero off-diagonal entries, we can localise the problem by permuting the matrix in such a way as to bring those entries as close as possible to the diagonal. Two diagonal entries are coupled if there is a non-zero where their row and column intersect. Each group of coupled entries is called a Jordan block. 

Suppose that there is a single off-diagonal non-zero element. Then we can 

Thus where off-diagonals appear, they link equal values on the diagonal. In general such a block can be larger than 2x2, but all the diagonal values will be equal. 

2 Effect of a Jordan block on the multiplication of a (generalised) eigenvector by the matrix. 

We can focus on the 2x2 case, because eigenvectors which are not associated with the block have a zero inner product with the block s rows. 

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If A has repeated eigenvalues (multiple roots of the characteristic polynomial), the result, again from introductory linear algebra, is the Jordan canonical form. Briefly, the transformation matrix P now needs a set of generalized eigenvectors, and the transformed matrix J = P 1 AP is made of Jordan blocks for each of the repeated eigenvalues. For example, if matrix A has three repealed eigenvalues A,j, the transformed matrix should appear as... 

In this tribute and memorial to Per-Olov Lowdin we discuss and review the extension of Quantum Mechanics to so-called open dissipative systems via complex deformation techniques of both Hamiltonian and Liouvillian dynamics. The review also covers briefly the emergence of time scales, the definition of the quasibosonic pair entropy as well as the precise quantization relation between the temperature and the phenomenological relaxation time. The issue of microscopic selforganization is approached through the formation of certain units identified as classical Jordan blocks appearing naturally in the generalised dynamical picture. 

For convenience we will make a simple demonstration of how to transform a 2x2 matrix problem to complex symmetric form. In so doing we will also recognise the appearence of a Jordan block off the real axis as an immediate consequence of the generalisation. The example referred to is treated in some detail in Ref. [15], where in addition to the presence of complex eigenvalues one also demonstrates the crossing relations on and off the real axis. The Hamiltonian... 

Thus we leam three things 1) the non-crossing rule is not obeyed in the present picture of unstable resonance states, 2) complex resonances may appear on the real axis and 3) unphysical states may appear as solutions to the secular equation. Thus avoided crossings in standard molecular dynamics are accompanied by branch points in the complex plane corresponding to Jordan blocks in the classical canonical form of the associated matrix representation of the actual operator. 

As we have seen in the previous section, Jordan blocks appear easily in an ever so slight non-Hermitean extension of Quantum Mechanics [19]. The success in both atomic as well as molecular applications has also been noted [20-22]. Non-trivial extensions from the Hamiltonian to the Liouville picture was moreover soon realised [23]. 

To give a simple demonstration of the consequences of an existing Jordan block in the associated dynamics we will consider the frequency operator... 

In Eq. (15) we have introduced a fast time scale or frequency co0 and a life time x. I is the unit and J the operator corresponding to a Jordan block representation. 

Hence, if we can find a relation between e k, p and r so that Eq. (37) assumes the Jordan block form (i.e. proportional to Q or J) one of the consequences for the dynamics is the emergence of an dramatically increased lifetime as indicated in our discussion in section 3. Hence if (see [10] for more details)... 

To belabor this point, let us consider in more detail a simple case, Refs. [78, 79], where the bound states of the Coulomb potential, through successive switching of a short-range barrier potential, becomes associated with resonances in the continuum. The simplicity of the problem demonstrates that resonances have decisively bound state properties, yields insights into the curve-crossing problem, and displays the tolerance of Jordan blocks. The potential has the form... 

The second point relates with emerging degeneracies of so-called Jordan block type, a nightmare in matrix theory but in our case, at the same time a "blessing in disguise."... 

In this appendix, we will derive a complex symmetric form for the Jordan block, see Eq. (E.l). We will also learn how such a degenerate representation may emerge in a realistic situation where the map reflects the property of an open (dissipative) structure. A general proof of the theorem, see below, was given already by Gantmacher [105] in 1959, but the theorem seems to be seldom mentioned. Here we will give an alternative proof, which also provides an explicit result that is also suggestive in connection with physical applications. 

First we observe that any matrix is similar to a block diagonal matrix, where the sub-matrices along the main diagonal are Jordan blocks. It is thus sufficient to prove that any Jordan block can be transformed to a complex symmetric matrix. In passing we note that any matrix with distinct eigenvalues can be brought to diagonal form by a similarity transformation. The key study therefore relates to XI + J (0), where 1 is the n-dimensional unit matrix and... 

This explicit construction proves the theorem. What remains is to work out the actual form the Jordan block Q... 

As already stated, the transformation B is not unique. Nonetheless it is interesting, with the aforementioned intermezzo as background, to note that the result (E.13), to be demonstrated below, i.e., em(k+l 2)/n(Ski — 1) carries a crucial semblance with r 2) in Eq. (E.5). In fact T(2), see more below, relates the complex symmetric representation of the Jordan block through thermaliza-tion, and furthermore the matrix B can also be used to diagonalize the latter. Employing the transformation H)B = g) = g, g2, g ) Eq. (E.5) writes... 

E. Brandas, Complex Symmetry, Jordan Blocks and Microscopic Selforganization An Examination of the Limits of Quantum Theory Based on Nonself-adjoint Extensions with Illustrations from Chemistry and Physics, in N. Russo, V. Ya. Antonchenko, E. Kryachko (Eds.), Self-Organization of Molecular Systems From Molecules and Clusters to Nanotubes and Proteins, NATO Science for Peace and Security Series A Chemistry and Biology, Springer Science+Business Media B.V., Dordrecht, 2009, p. 49. 

As shown previously analogous equations can be derived in a statistical framework both for localized fermions in a specific pairing mode and/or for bosons subject to a quantum transport environment [7]. The second interconnection regarding the relevance of the basis f is related to the fact that a transformation of form (20) connects canonical Jordan blocks to convenient complex symmetric forms. This will not be explicitly discussed and analysed here except pointing out the possible relationship between temperature scales and Jordan block formation by thermal correlations (see e.g. [7-9,14], for more details). 

Returning now to the zero mass case, we note that we have a Jordan block situation irrespective of the value of r in Eq. (35). For a non-zero mass particle Eq. (39) determines the correct Schwarzschild radius. Hence consistency between Eqs. (35) and (36) and Eq. (39) requires... 

We have previously observed that the repeated use of Eqs. (12) and (13), after diagonalization, followed by a new degenerate Jordan block state, yields e-doubling. The expansion of the universe, i.e. Hubble s law, possibly due to a distant hidden black hole-like structure, could in principle lead to amplified contractions of time and length dimensions. From Eq. (34), i.e. reassigning p —> p = p( 1 — k(t)) follows the interpretation that the momentum will be r-dependent. 

There is finally the possibility of decay-like leakages between the particle-antiparticle spaces, and further that there could be an overall escape out of the presently defined spaces. If so associated Jordan blocks naturally appearing would decelerate this decay via the polynomial delay mechanism described earlier [7-10] with implications to subject matters like problems related to size of the cosmological constant. Also, the account given here should consider a more general decomposition of pz into curvilinear coordinates (cf. Eq. (30)) in order to yield a more appropriate analysis (see e.g. [22] and references therein). 

E.J. Brandas, Are Jordan blocks necessary for the interpretation of dynamical processes in nature Adv. Quant. Chem. 47 (2004) 93-106. 

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