# Tagged Questions

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### Proving Product of Transition Matrices is again a Transition Matrix.

Let $P = [p_{ij}]$ be an $n\times n$ transition matrix for an $n$-state markov chain. How do you prove that $P^2$, or even better, that $P^n$ is again a transition matrix? My approach leaves me ...
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### Applying transition matrix to a probability vector seems to ruin its normalization

I had a little bit about stochastic processes during my "Statistical Physics" course and on my exam I got a problem with a Markov chain. My solution seems to be without computational mistakes (checked ...
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### Proof that steady state is not affected by initial distribution in Markov chain.

I was following a proof provided in Gilbert Strang's book "Introduction to Linear Algebra". And I am confused by one step of the proof. Suppose we have a $n$ by $n$ stochastic matrix $A$, where all ...
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### PRobability Markov chain, system of equations

I'm looking for techniques or tricks to solve a system of linear equations you get where you want to find the limiting probabilities. The system is this: $\pi_0 = 0.7\pi_0 + 0.2\pi_1 + 0.1\pi_2$ ( ...
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### How is the Chapman-Kolmogorov Equation not a fancy name for matrix multiplication?

The Chapman-Kolmogorov Equation: $$p^{m+n}(i,j)=\sum_kp^m(i,k)p^n(k,j)$$ Matrix Multiplication (with $[A]_{i,j}=a_{i,j}$ where $A$ is a linear map "" for B) $$[AB]_{i,j}=\sum_ka_{i,k}b_{k,j}$$ In ...
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### Markov Chain with Normal Transition Matrix

Consider a (sub)-stochastic matrix $P$, and the associated Markov chain $X$ with \begin{align*} \mathbf P [X_n =y | X_0 = x] = P_{xy}^n. \end{align*} Suppose we have the condition $P^T P = P P^T$, ...
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### 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 ...
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### 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 ...
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### Example of a Markov chain transition matrix that is not diagonalizable?

It is well-known that every detailed-balance Markov chain has a diagonalizable transition matrix. I am looking for an example of a Markov chain whose transition matrix is not diagonalizable. That is: ...
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### Perron-Frobenius theorem

In the proof of the Perron-Frobenius theorem why can we take a strictly positive eigenvector corresponding to the eigenvalue $1$? Before that, why can we even take a non-negative eigenvector? Books ...
In a certain text on Perron-Frobenius theory, it is postulated that every irreducible nonnegative matrix with period $1$ is primitive and this proposition is said to be obvious. However, when I tried ...