In this paper, we present a different approach to identifying metastable states of a Markov chain: we find a permutation of a given stochastic transition matrix of a Markov chain, such that the resulting matrix is block diagonally dominant.
We call the matrix T (strictly) diagonally dominant if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all i = 1,.
However, as will be shown momentarily, in our case H may not be positive definite in this entire region, and it need not be strictly diagonally dominant, although several numerical experiments seem to indicate that the equilibrium is indeed stable and unique.
To end this section, we analyse a specific example for which it can be proved that the equilibrium position is stable, even though the Hessian is not strictly diagonally dominant at this position.
In these early papers all the matrices involved had to be diagonally dominant, but that is irrelevant for our paper.
GREMBAN, Combinatorial Preconditioners for Sparse, Symmetric, Diagonally Dominant Linear Systems, PhD thesis, School of Computer Science, Carnegie Mellon University, Oct.
Our examples include diagonally dominant matrices by rows and columns and their inverses, Stieljes matrices and M-matrices diagonally dominant by rows or columns.
In Section 2 we also prove that if either a nonsingular matrix or its inverse is diagonally dominant by rows and columns, then we can assure an LDU-decomposition of A (without row or column exchanges) with L and U diagonally dominant by columns and rows, respectively.
2 is only weakly diagonally dominant and the entries of the inverse of A decay much more slowly with respect to those of the matrix in Figure 3.
Let A be a normalized symmetric positive definite diagonally dominant matrix, and let [alpha]E, [alpha] [member of] [C.
n x 2], and given any nonempty subset S of N, then A is an S-strictly diagonally dominant matrix if
i](A) ai,i I > ri (A) (all i [member of] N), and this is just the familiar statement that A is strictly diagonally dominant.