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.
217 questions
6 votes
1 answer
101 views
Which similarity transformations preserve stochasticity of a matrix.
An $n\times n$ real matrix is called stochastic (aka Markov) if it is entry-wise non-negative and it's rows sum to 1. Let $\mathcal{M}_n^+$ be the set of stochastic matrices. I am interested in the ...
- 312
3 votes
1 answer
283 views
Existence of a kernel-preserving stochastic factorization of a column-stochastic matrix
I'm working on a matrix factorization problem and would appreciate insights on the following conjecture: $\forall y,z \in \mathbb{N}$ such that $y > z$, there does not exist column-stochastic $\...
- 85
2 votes
1 answer
67 views
Transformation of matrices
Suppose $M_1$ and $M_2$ are both row-stochastic matrices, that is, every entry is non-negative and the sum of entries in the same row is 1. One can interpret the rows as states while the columns as ...
- 1,226
1 vote
0 answers
50 views
Given a random matrix A, does this inequality hold $\rho \left(E[A \otimes A]\right) \leq \rho \left(E[AA^T ]\right)$
I am currently working on a problem related to a stochastic dynamical system defined as: $x^+ = A x$, where $A$ is a random matrix supported on $\mathbb{R}^{n \times n}$, I am so far assuming that the ...
- 11
2 votes
1 answer
118 views
Do non-diagonalizable Markov matrices have a limit?
Considering the definition of a Markov matrix or a stochastic matrix as the matrices such that all their columns sum exactly 1 and all of their entries are nonnegative, I know that if the Markov ...
11 votes
3 answers
713 views
If a stochastic matrix has unit permanent, is it a permutation matrix?
In this question, a stochastic square matrix is a real square matrix where all the rows sum up to $1$ and all the entries are between $0$ and $1$. Permutation matrices are examples of stochastic ...
- 349
1 vote
1 answer
84 views
Norm inequality for a column stochastic matrix whose columns are projected onto a subspace orthogonal to the Perron eigenvector
Please prove or disprove the following conjecture. Conjecture. Let $B \in \mathbb{R}^{n \times n}$ be a column stochastic (also known as left stochastic) matrix, and let $ \pi \in \mathbb{R}^n$ be ...
3 votes
1 answer
122 views
Eigenvectors of reversible stochastic matrix
Let $P$ be the stochastic matrix associated with a reversible Markov chain and $\pi$ its stationary distribution. Then it can be seen that $DPD^{-1}$ where $D=\text{diag}(\sqrt{\pi(x)})$ is symmetric ...
- 375
-1 votes
1 answer
69 views
Starting in state $1$ of a Markov chain, what is the probability of absorption at state $3$? [closed]
For the following transition probability matrix $$ P = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0.3 & 0.1 & 0.3 & 0.3 \\ 0.4 & 0.1 & 0.4 & 0....
- 9
1 vote
1 answer
168 views
Infinite product of doubly stochastic matrices
Suppose ${\bf A}_1, {\bf A}_2, \dots$ is a sequence of doubly stochastic $n \times n$ matrices, i.e., all rows and columns of each $A_k$ sum to $1$. Suppose further that all entries of each ${\bf A}_k$...
user249238
0 votes
0 answers
47 views
Name of this technique for getting the eigenvector of a Markov Matrix
I have the following Markov matrix $$ M = \begin{bmatrix} 1-a & 1-b & 1-c & 1-d \\ a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & c & ...
- 1
4 votes
1 answer
280 views
On powers of a Markov matrix
Recently, I came across a very nice question from the 1985 AIME: Let $A$, $B$, $C$ and $D$ be the vertices of a regular tetrahedron, each of whose edges measures $1$ meter. A bug, starting from ...
- 1,477
0 votes
0 answers
66 views
Inverse of regular stochastic matrix
What can we say about the invertibility of a regular stochastic matrix? Definitions (based on Howard Anton, Elementary Linear Algebra, 12th ed.) A vector with nonnegative entries that add up to 1 is ...
- 1
4 votes
0 answers
244 views
A generalization of Sinkhorn's theorem
Let $A$ be an entry-wise strictly positive matrix. Sinkhorn's theorem assures that there exist strictly positive diagonal matrices (unique up to scaling) $D_1, D_2$ such that $D_1 A D_2$ is doubly ...
- 1,244
1 vote
1 answer
65 views
Similarity between stochastic matrices
Suppose I have two stochastic matrices $X,Y \in \mathbb{R}^{n \times n}$, ie. they are non-negative and the rows sum to 1. Does there exist $T \in \mathbb{R}^{n \times n}$ such that $Y = TX$ or $Y = ...
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