Jordan canonical form

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 For example, if matrix A has three repealed eigenvalues A,j, the transformed matrix should appear as... 

This reduction of A to diagonal form is unique, except for the order in which the eigenvalues occur on the diagonal. When the roots are not all distinct, it may not be possible to convert A to diagonal form. However, A may then be reduced to Jordan canonical form in which the eigenvalues occur on the diagonal, with the position immediately above each eigenvalue occupied by unity or zero and with a zero everywhere else. 

Finally, the canonical form T MT expressed in the lemma is just the real Jordan canonical form of M. ... 

The Jordan canonical form of a matrix is best defined in terms of elementary matrices Je which are matrices with entries — X, along the principal diagonal, entries 1 on the diagonal next below the principal diagonal and all other entries zero. For example. 

Let us now show that the Jordan canonical form is similar to a matrix in the canonical form N [Eq. (183)]. The elementary Jordan matrices are transformed into the required form by... 

When TO = 1, matrices in the Jordan canonical form, we have... 

At the end of the Conference on Irreversible Quantum Dynamics in Trieste, in August 2002, Arno Bohm addressed the question of the physical interpretation and the occurrence in Nature of processes related to the appearance of Jordan blocks. He ended the presentation by asking, where are all the Jordan blocks The question is not new as the possibility of Jordan canonical forms in an extended quantum dynamical picture has been noticed many times in the past decades, see Refs. [1,2] and references therein. The topic was again resumed at the QSCP workshop in Bratislava a month later. 

G.H.Golub, J.H.Wilkinson El-conditioned Eigensystems and Computation of the Jordan Canonical Form. SIAM Review 18, pp. 578-619 (1976)... 

In this sub-case, the ODE system matrix A is not equivalent to a diagonal matrix D, but it is equivalent to a triangular matrix / (the Jordan canonical form). More precisely, there exists a matrix P (it is important to note that in this case not all the columns of P are eigenvectors) such that P A P = and as a consequence, the ODE system X = A X can be easily converted in the following one Y = JY by a simple change of variables X = PY. ... 

In this new sub-case, the corresponding Jordan canonical form has the following form ... 

In this last case the Jordan canonical form is / =... 

This article describes a combination of chemical and mathematical modelling applied to the adsorption of Carbon Dioxide on Platinum surfaces, but a similar procedure can be applied to any chemical or electrochemical mechanism involving unimolecular reactions. Moreover, mathematical theorems about eigenvalues, eigenvectors, diagonalization, Jordan canonical forms, etc., and chemical laws, particularly Lavoisier s law of mass conservation can be combined to solve inverse causation and stability problems. 

Proof As we already saw in Case 2, the Jordan canonical form plays an important role. It can be obtained by an appropriate similarity transformation applied to the system matrices. Here, a pair of matrices (E, A) must be transformed simultaneously. By this similarity transformation the matrix pencil can be transformed into the so-called Kronecker canonical form pE — A, where both, E and A, are block diagonal matrices consisting of Jordan blocks. 

In the next and last transformation we use the fact that the blocks —IfjiN) N and have their own Jordan canonical forms fj,N) NTjsi = N... 

In general this transformation step cannot be performed in a numerical robust way, as the process demands the transformation of matrices to its Jordan canonical form. This canonical form cannot be established by means of stable algorithms. In contrast, for mechanical systems the matrix E A can be given explicitly [SFR93]. This will be shown in the next section. 

The first step in the calculation of the matrix exponential can be (Prasolov 1994) the decomposition of matrix Jo to its Jordan canonical form J using the invertible matrix P ... 

As already mentioned the informity rule prompts several consequences one being the emergence of so-called Jordan blocks or exceptional points. Although belonging to standard practise in linear algebra formulations we will proffer some extra time to this concept. In addition to demonstrate its simple nature we will also establish a simple complex symmetric form not previously obtained, see e.g. Refs. [11,14, 21,22]. Let us start with the 2 x 2 case, where it is easy to demonstrate that the Jordan canonical form / and the complex symmetric form Q are unitarily connected through the transformation B, i.e. 

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