1
vote
0answers
182 views

Closed-form solution to a system of linear equations

Consider the following $n \times n$ matrix with a particularly nice structure: \begin{equation}\mathbf{P}=\begin{pmatrix} 0 & 0& \dots&0 & 0 &1\\ 0 & 0& \dots&0 & ...
12
votes
4answers
544 views

Eigenvectors of a particular transition matrix

I am considering a Markov chain with $n$ states with a particularly nice structure. The transition matrix is as follows: \begin{equation}\mathbf{P}=\begin{pmatrix} 0 & 0& \dots&0 & 0 ...
6
votes
1answer
148 views

A family of skew-symmetric matrices corresponding to cycles in graphs

When investigating loops in Markov chains I ran into the following observation. A cycle in a graph $G$ with $n$ vertices may be represented by a matrix $\Gamma \in \mathbb R^{n \times n}$ having the ...
5
votes
1answer
641 views

Efficient computation of Markov chain transition probability matrix

Consider a continuous Markov chain $X = (X_t)$ on a finite state space and let $Q$ be the (given) transition rate matrix. This matrix is very sparse, with non-zero values on 3 diagonals only (so from ...
8
votes
1answer
501 views

Bounds on $||P^{k+1} - P^k||$ for $n$ by $n$ stochastic matrix $P$ with trace $n-1$ and integer $k>>n$.

The problem: We have a $n$-state Markov chain with arbitrary initial distribution and transition matrix $P$ that is arbitrary except that we know that $P$ has trace $n-1$. Of course $P$ is also a ...
1
vote
2answers
451 views

Markov chain convergence problem.

Consider a markov chain matrix P of size n x n (n states). P is known to be: 1- there are at least two absorbent states. one of them is denoted by null. (thus, we have that P_null,null = 1) 2- For ...
-2
votes
3answers
650 views

Convergence of a markov matrix

Consider a markov chain matrix P of size n x n (n states). P is known to be: 1- Not irreducible (i.e. there exist at least a pair of states i, j such that we cannot go from i to j) 2- Not all ...
3
votes
1answer
348 views

(Stochastic) matrix for which a stochastic matrix logarithm exists?

I think this is basically the inverse question of Matrices whose exponential is stochastic. i.e. what are sufficient conditions on the matrix representation of an evolution operator of a (finite) ...
12
votes
0answers
703 views

Diagonalizing some matrices arising from Fourier transform on $S_n$.

Consider the function $f$ on $S_n$ which equals $1/n$ on all adjacent transpositions $(i,i+1)$, where we let $n+1 = 1$, and $0$ otherwise, and its Fourier transform $\hat{f}(\rho)$ evaluated at the ...