Block-triangular matrices

For the important case in which the matrices Bij = 0 for jf > i, it follows that the matrix (7 — (/ tj2)Q) is a block triangular matrix, and in the case of one space variable where the diagonal blocks are tridiagonal, (7 — (A /2)Q) can be directly inverted. For two or more space variables, the process of solving (7.5) for large numbers of mesh points would involve again inner iterations. 

A particular example of information which can be exploited is when the matrix M is of block triangular structure. 

Thus the matrix C of eigencolumns has the same block-triangular structure as M. 

Reduce the matrix A to the block diagonal form or to the block triangular form with the maximum number of blocks (policies 1 and 2 amount to computing all the connected components or all the strongly connected components, respectively, of the associated graph G... 

Calculating the matrix elements of the Hamiltonian in this basis set gives a sparse, real, and symmetric M(N) x M(N) matrix at order N. By systematically increasing the order N, one obtained the lowest two eigenvalues at different basis lengths M(N). For example, M(N) = 946 and 20,336 at N = 20 and 60, respectively [11]. The symmetric matrix is represented in a sparse row-wise format [140] and then reordered [141] before triangularizations. The Lanczos method [142] of block-renormalization procedure was employed. 

The next simplification of the calculations is by exploiting the block structure of these matrices. We use the property that if a matrix is block lower triangular, its eigenvalues are those of the diagonal blocks. 

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... 

The transition intensity matrix has the block form (1) but the matrices Qs and Qg are upper triangular. Similarly, the embedded Markov chain will have the transition probability matrix (7). 

The above considerations have also been used to construct a simple method for the iterative solution of the linear block system (2.9). Let A denote the Jacobian approximation in (2.9) and A the associated matrix with Qy replaced by 0. Then the system Ax = h can be solved easily because of its nearly upper triangular block structure. On this basis, we constructed a fixed point iteration which is known to converge with contraction rate not greater than the spectral radius p(I — A A), Obviously, since y = 0 in the starting point xo, we will have p < 1 in some neighborhood, which can be monitored. 

To infer from the allowed form of t to the allowed form of D, we note that it can be shown that, if A has the upper triangular block structure of Figure 11.7, then L in the LU decomposition A = LU will have the same structure. Hence, if we require D to have the stmcture of Figure 11.7, then t in (11.9.25) will have the same structure. We have thus identified the structure of D that obeys the RAS constraints. Note that the matrix in Figure 11.7 is not sufficiently flexible for a general transformation - for example, for transforming the RAS wave function to the natural-orbital basis. 

A matrix A is irreducible if there exists no permutation matrix P, such that PAP takes the block upper triangular form... 

As we demonstrate below for an example matrix, as oo, this sequence of matrices becomes of block upper triangular form,... 

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