Matrix Jordan form

From the quantization condition Eq. (F.10), see also Ref. [7], we that the thermalized matrix, Eq. (F.8) admits the Jordan form... 

In equation (17) the unitary matrix B connects the standard Jordan form J... 

From (1.55) we realize that the thermalized matrix in Eq. 1.54 assumes the Jordan form... 

SO that the matrix A in (12.1.1) is in the Jordan form and, moreover, the off-diagonal entries, if there are any, are sufficiently small. If d is small, then the function h in (12.1.1) is also sufficiently small elsewhere in Vq. Hence, the following estimate for the trajectories in Vo is valid ... 

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 For example, if matrix A has three repealed eigenvalues A,j, the transformed matrix should appear as... 

The classification procedure developed by Madron is based on the conversion, into the canonical form, of the matrix associated with the linear or linearized plant model equations. First a composed matrix, involving unmeasured and measured variables and a vector of constants, is formed. Then a Gauss-Jordan elimination, used for pivoting the columns belonging to the unmeasured quantities, is accomplished. In the next phase, the procedure applies the elimination to a resulting submatrix which contains measured variables. By rearranging the rows and columns of the macro-matrix,... 

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. 

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... 

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)... 

If, finally, P is the permutation matrix built up in block-diagonal form from all the submatrices Pt associated with the various Jordan blocks, one... 

A multireaction version of Eq. (2.2-2) is obtainable by continuing the elimination above as well as below the pivotal elements this is the Gauss-Jordan algorithm of Section A.3. The final nonzero rows form a matrix a, here given by... 

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 the matrix is reduced to echelon form by Gauss-Jordan elimination, the rank of the matrix can be shown to be equal to 3. With n = 5, the number of independent reactions is 5 - 3 = 2. Equation (4.575) requires that, for each of the two independent reactions,... 

This is a systematic procedure for carrying out the method of elimination. It is very similar to the Gauss-Jordan method for finding the inverse of a matrix, described in Chapter 9. If the set of equations is written in the vector form... 

E) can be easily obtained from Eq. (25) by diagonalization of a smalldimensional non-Hermitian effective Hamiltonian. This is not true if the effective Hamiltonian has the form of a Jordan block (see p. 475 in Ref. [2]). A simple example of Jordan structure is the matrix representation (30) in Section 2.2.2 when = 4A. In this nonphysical case the two eigenvalues coalesce at the value -2iA. 

In Eq. (1.12), we have chosen a momentum p in an arbitrary direction with the mass consistently given by pic. We also note another detail. The operator matrix and its representation must, as we have demonstrated above, have a complex conjugate in the bra-position. However, since we here encounter a degeneracy with the Segr6 characteristic equal to two, we have attained a so-caUed Jordan block in disguise . To display the more familiar canonical (triangular) form of the description, we... 

An efficient way to calculate the inverse of a square matrix A of order n is (a) Place the nth-order unit matrix I at the right of the matrix A to form an n-row, 2n-colunm array, which we denote by (AjI). (b) Perform Gauss-Jordan elimination on the rows of (Ajl) so as to reduce the A portion of (All) to the unit matrix. At the end of this process, the array will have the form (liB). Ilie matrix B is A . (If A does not exist, it will be impossible to reduce the A portion of the array to I.) Use this procedure to find the inverse of the matrix in Problem 8.42. 

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. ... 

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