# Questions tagged [stochastic-matrices]

A right stochastic matrix is a nonnegative square matrix with each row summing to 1. A left stochastic matrix is a nonnegative square matrix with each column summing to 1. A doubly stochastic matrix is both right stochastic and left stochastic.

139 questions
Filter by
Sorted by
Tagged with
70 views

### Confusion in one statement related to the Birkhoff polytope

I know that the set of doubly stochastic matrices $(\Omega(n))$ form a polyhedron. I only know about polyhedron is that it is a $3$-dimensional shape with flat polygonal faces, straight edges and ...
1 vote
57 views

### State Transition Matrix

A similar question to this post, but now row sums are zeros. I don't think the suggeted approach in the comment works anymore. Given $$\dot x = A(t)x(t)$$ where $A(t)$ is a Metzler matrix whose row ...
60 views

### State Transition Matrix of a time varying linear dynamical system

Given $$\dot x = A(t)x(t)$$ where $A(t)$ is a Metzler matrix whose column sums are zeros at all time $t$, prove that the state transition matrix of the system is a left (or right) stochastic matrix ...
1 vote
72 views

### Prove that a matrix can be written as a sum of permutation matrices

Given a square matrix $A$ of size $n$ whose entries are non-negative integers and where the sum of each column and row is equal to $k$, prove that $A$ can be written as a sum of $k$ permutation ...
79 views

• 11
46 views

### For reversible Markov chains larger off diagonal elements imply smaller eigenvalues

The following is claimed in Remark 3.1 in this paper (page 6). Let $M$ and $M'$ be $\pi$-reversible Markov chains on $X$ with eigenvalues $1=\lambda_0 \ge \lambda_1 \ge \cdots \ge \lambda_{|X|-1}$ and ...
15 views

### Are there any good results characterizing the difference in stochastic matrices?

We frequently use stochastic matrices $P$ in various settings (for example, MDPs). In practice, we estimate stochastic matrices using observed samples and arrive at an estimate $P_0$. Usually there is ...
• 959
1 vote
55 views

### Eigenvectors of Stochastic Matrix

Let $N\in\mathbb{N}$. Define the matrix $Q\in\mathbb{R}^{(N+1)\times(N+1)}$ by $$Q_{ij}=\binom{N}{j}\Big(\frac{i}{N}\Big)^j\Big(1-\frac{i}{N}\Big)^{N-j}$$ for all $0\le i,j\le N$. The sum of each row ...
• 364
1 vote
95 views

### Power of Stochastic Matrix

Given the stochastic matrix $$Q = \begin{pmatrix}0&2/3&1/3\\1/3&0&2/3\\2/3&1/3&0\\\end{pmatrix}\in \mathbb{R}^{3\times3}$$ I wish to compute $Q_{1,1}^n$ (the entry in the first ...
• 364
89 views

### Finding a pair of commutative stochastic $3 \times 3$ matrices

Could you please give me an example of a pair $\left(A,B\right)$ of commutative stochastic $3 \times 3$ matrices with real entries given $A \neq B$ and excluding the identity matrix, symmetric ...
31 views

### Why doesn't detailed balance always imply a uniform distribution for Markov Chains?

Consider a state (row) vector $\{\pi_i\}$ and a right-stochastic matrix $P_{ij}$. To calculate the next state vector $\pi'$ you can right-multiply by $P$: $$\pi'_i=\sum_j \pi_jP_{ji}$$ Detailed ...
42 views

### Can we approximate the linear sum assignment solution with bistochastic matrices?

This question was concerned with random bistochastic matrices and the answer pointed out an algorithm, that can "bistochastize" a starting matrix $\boldsymbol{X}$ (with positive elements ...
• 133
79 views

• 1,255
41 views

### How can I make sense of this Matrix inequality?

In https://tel.archives-ouvertes.fr/tel-02926037/document one page 35 the following Lemma is stated: Lemma I.5: Let $M \in (\Bbb R^+)^{N \times N}$ be a stochastic matrix. Then for any positive ...
• 1,255
39 views

### Interpretation of the Sinkhorn-Knopp algorithm applied to a (singly stochastic) transition matrix of a Markov process?

Say I have a discrete-time Markov process (and let's say discrete states too, for simplicity). If $\mathbf p_t$ is a vector of probabilities over states at time $t$, then the probability distribution ...
• 289
33 views

### Factors of a stochastic matrix [closed]

If $A B = C$ where matrix $C$ is stochastic (all entries are positive and all rows add to $1$) then is it necessary that both $A$ and $B$ be stochastic?
58 views

• 179
525 views

### Finding stable distribution of a Stochastic Matrix

so im currently looking at Markov diagrams and stochastic matrices and have become quite stuck on some parts of this question below and was looking for some help. Here is what I have done so far, For ...
• 349
22 views

### What's the application of doubly-stochastic matrices in engineering?

Today I learned the existence of such matrix . wolframe This is indeed a very interesting thing. I am wondering if there is any realworld application of such matrix. It seems that this matrix has ...
• 183
1 vote
88 views

### Conditions under which a row-stochastic Markov matrix has positive eigenvalues

Let $A \in \mathbb{R}_+^{n\times n}$ be the adjacency matrix of a weighted directed graph, i.e., $A$ is nonsymmetric and with nonnegative entries. Let $M = D^{-1}A$ be the row-stochastic Markov matrix ...
• 63
1 vote
855 views

### Proving a specific eigenvalue of a 6x6 Matrix.

Currently, I am looking at Markov diagrams and their associated transition matrices. I am trying to prove that the transition matrix has eigenvalue $\lambda=1$. I am aware that to find the eigenvalues ...
• 349
1k views

### Determinant of a 6x6 matrix

There's this part of my assignment which involves stochastic matrices and i've done most parts of it but there's one part which requires me to show that its eigenvalue is 1. The only way i can think ...
128 views

### Raising non-diagonalizable Markov transition matrix to the power of $n$

Consider an aperiodic and irreproducible Markov process with absorbing states. For each non-absorbing state, how can I numerically compute the likelihood of ending up in each absorbing state? Assume ...
• 123
301 views

• 135
53 views

246 views

### Convergence of Markov Chains

I have stumbled upon Markov chains respectively finite stochastic matrices (gladly it didn't hurt too much). There is this theorem on the existence of stationary distributions which is standard: Let P ...
1 vote
92 views

### Eigenvalues of Markov matrix

I was trying to prove that a Markov matrix always has an eigenvalue 1 but I seem to have proved that all eigenvalues of a Markov matrix are equal to one! Could anyone look at my work and point out the ...
• 461
1 vote
133 views

### Can we find a stochastic matrix $Q$ such that $M = \frac{1}{C} \sum_{i=1}^C Q^i$ where $M$ is a stochastic matrix?

Given a stochastic matrix $M$, can we always find another stochastic matrix $Q$ such that $M$ can be written by the mean of a geometric sum $M = \frac{1}{C} \sum_{i=1}^C Q^i$ for any $C > 1$? My ...
• 125
1 vote
143 views

### Uniform distribution on the space of stochastic matrices

Does there exist any uniform distribution on the space of (right) stochastic matrices (a real square matrix, with each row summing to 1)? If so, is it unique? Besides the uniform distribution, does ...
• 125
Consider stochastic matrix $P$, it means that the sum of elements in row is equal $1$. Matrix $P$ is function of $p\in\left(0,1\right)$, it means that P=\begin{pmatrix}f_1(p) & f_2(p) & f_3(...