0
votes
1answer
25 views

Control principal eigenvector of a row stochastic matrix

I am just trying to consider the classical discrete-time Markov Chain problem. Consider the transition matrix P, which transforms state vector $x(k)$ to $x(k+1)$, satisfying: $x(k+1)$ = $P*x(k)$ It ...
-1
votes
0answers
15 views

Does a transition matrix has only real positive eigenvalues? [duplicate]

I am analyzing an ergodic Markov Chain of an $N\times N$ lattice grid and I have written down my Transition Matrix, i.e. all values are between $0$ and $1$ and the elements of each row sum up to $1$. ...
0
votes
0answers
16 views

The second eigenvalue of a reducible stochastic matrix

The magnitude of the second dominant eigenvalue of a reducible matrix, as I know, is supposed to be 1, why it's not the case for this matrix : $$ \begin{matrix} 0 & 1 & 0 ...
5
votes
1answer
168 views

Generalization of the Jordan form for infinite matrices

Under what conditions is it the case that for a matrix $M$ whose rows and columns are indexed by a countably infinite set $S$ one has a Hamel basis consisting of generalized eigenvectors (i.e. $v \in ...
0
votes
1answer
63 views

Convergence rate of PageRank, the problem when the second eigenvalue is complex

As far as I know the Google matrix used to calculate the PageRank is not symetric, that means that some eigenvalues can be complex, furthermore, we know that the second eigenvalue is equal to the ...
0
votes
1answer
31 views

Why does markov chains power method converge at the rate of |λ_2/λ_1 |

I'm doing some researches on Markov Chains, and every time I meet this statement, that The rate of convergence of the power method is given by |λ_2/λ_1 |^k→0, when k→inf. And where λ_1 and λ_2 are ...
2
votes
2answers
56 views

Transition Matrix eigenvalues constraints

I have a Transition Matrix, i.e. a matrix whose items are bounded between 0 and 1 and either rows or columns sum to one. I would like to know if it is possible that in any such matrices the ...
0
votes
1answer
24 views

How many iterations are generally required when using the power iteration method?

Suppose I have an n x n matrix and I want to find the dominant eigenvalue and its associated eigenvector. Given these dimensions, what is the minimum number of iterations of the power iteration ...
2
votes
1answer
114 views

Proof that Markov Chains converges to the stationary distribution

Let $P$ is a transition matrix of a Markov Chain, which is irreducible, aperiodic and lets assume $\pi$ is its stationary distribution: $\pi = \pi P$. Does anyone knows the proof for the following ...
1
vote
1answer
94 views

Markov Chain Solution Eigenvalue

I am having trouble understanding how to solve for the state vector at time $t$ for a markov chain using matrix algebra. I have the following Markov Transition Intensity Matrix, for the states A, N, ...
1
vote
1answer
123 views

Steady State Markov Chain

I was reading http://www.ams.org/bookstore/pspdf/mbk-58-prev.pdf and going through the first example for the frog jumping between the lily pads. I'm interested in find the steady-state probability for ...
1
vote
1answer
202 views

Eigenvector of transition matrix for Markov chain

Why is the only eigenvector of the transition matrix for an irreducible Markov chain with eigenvalue $= 1$ the eigenvector with all ones?
0
votes
1answer
73 views

Bounding the smallest eigenvalue of an ergodic Markov Chain

I am trying to prove that the smallest eigenvalue of an ergodic Markov chain is greater than -1. Can we do that using proof by contradiction, i.e. assuming the smallest eigenvalue being -1, etc.? The ...
14
votes
1answer
447 views

Eigenvalues for $3\times 3$ stochastic matrices

This is a plot of the non-real eigenvalues of 10000 randomly generated $3\times3$ stochastic matrices. It's pretty clear that they lie in the convex hull of the three cube roots of unity. The ...
2
votes
1answer
369 views

Eigenvalues of a infinitesimal generator matrix

Consider a Markov process on a finite state space $S$, whose dynamic is determined by a certain infinitesimal generator $Q$ (that is a matrix in this case) and an initial distribution $m$. 1) Is ...
0
votes
2answers
2k views

What is the eigenvalue of stochastic matrix?

I have a stochastic matrix $A \in R^{n \times n}$ whose sum of the entries in each row is 1. When I found out the eigenvalues and eigenvectors for this stochastic matrix, it always happens that one of ...