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.
Questions tagged [stochastic-matrices]
217 questions
71
votes
7 answers
Proof that the largest eigenvalue of a stochastic matrix is $1$
The largest eigenvalue of a stochastic matrix (i.e. a matrix whose entries are positive and whose rows add up to $1$) is $1$.
Wikipedia marks this as a special case of the Perron-Frobenius theorem, but I wonder if there is a simpler (more direct)…
koletenbert
- 4,150
22
votes
2 answers
Eigenvalues for $3\times 3$ stochastic matrices
This is a plot of the non-real eigenvalues of $10^4$ 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 boundary on the left hand side is easy to explain. If…
user940
18
votes
3 answers
How do I generate doubly-stochastic matrices uniform randomly?
A doubly-stochastic matrix is an $n \times n$ matrix $P$ such that
$$ \sum_{i=1}^n p_{ij} = \sum_{j=1}^n p_{ij} = 1 $$
where $p_{ij}\ge 0$. Can someone please suggest an algorithm for generating these matrices uniform randomly?
Henry B.
- 2,058
15
votes
1 answer
Is there any significance to this "doubly stochastic matrix" with both a discrete and continuous index?
This is just idle curiosity. Consider the function $(\lambda, n) \mapsto e^{-\lambda} \frac{\lambda^n}{n!}$, where $\lambda \in \mathbb{R}_{\ge 0}$ is a nonnegative real parameter and $n \in \mathbb{Z}_{\ge 0}$ is a nonnegative integer parameter.…
Qiaochu Yuan
- 468,795
13
votes
2 answers
What's the algorithm of finding the convex combination of permutation matrices for a doubly stochastic matrix?
According to Birkhoff, $n$-by-$n$ stochastic matrices form a convex polytope whose extreme points are precisely the permutation matrices. It implies that any doubly stochastic matrix can be written as a convex combination of finitely many…
xzhu
- 4,303
12
votes
1 answer
No solutions to a matrix inequality?
If $A$ is a stochastic matrix, then $A$ is entry-wise nonnegative and $Ae = e$, i.e., $(1,e)$ is a right eigenpair for $A$.
Is it true that there exists a vector $b$ such that
$$(A - I)x \geq b$$
has no solutions in $x$? If so, is there a simple…
Mike Spivey
- 56,818
11
votes
3 answers
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 square matrices, for which the permanent
$$…
Urh
- 349
9
votes
3 answers
Cesàro limit of a stochastic matrix
Let $A$ be a stochastic matrix. Then
\begin{align*}
\lim_{t \rightarrow\infty} A^t
\end{align*}
may not exist. For example:
\begin{align*}
A &= \begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix} \\
A^{2t} &= I \\
A^{2t+1} &= A
\end{align*}
Now define the…
user76284
- 6,408
8
votes
1 answer
When are the inverses of stochastic matrices also stochastic matrices?
A stochastic matrix, with elements $\in[0,1]$ and rows summing to 1 are known to have one eigenvalue 1 (stationary distribution) and the rest of lower magnitude. However I don't know about many results regarding their inverses.
In which cases is…
mathreadler
- 26,534
8
votes
2 answers
Sinkhorn theorem for doubly stochastic matrices
I was reading something about doubly stochastic matrices and got stuck while reading the original proof of the uniqueness part of the Sinkhorn theorem. I'm not able to understand the logic. Could someone help me in figuring it out? I state below the…
alecsphys
- 103
7
votes
4 answers
If $B = x(xI-A)^{-1}$ for a generator matrix $A$, then $B-B^2$ has positive diagonal elements
Let $A$ be the generator matrix of a continuous-time Markov chain. This means that $A$ has positive off-diagonal elements $A_{ij} > 0$, $i \ne j$, and row sums $\sum_j A_{ij}$ equal to $0$. For example, $A$ could be
$$
A = \left(
\begin{matrix}
-7 &…
user133281
- 16,341
7
votes
2 answers
Proving or disproving product of two stochastic matrices is stochastic
Let $P$ and $Q$ be two stochastic matrices. Does the product $PQ$ have to be stochastic? Prove or disprove.
What Im thinking is that since matrix multiplication is only defined for two matrices $A$ and $B$ where $A$ has the same amount of columns…
Pame
- 1,073
7
votes
0 answers
Birkhoff representation of a stochastic matrix
From the Birkhoff theorem, it is known that every doubly stochastic matrix can be written as a convex combination of permutation matrices, although this representation might not be unique.
Assume that a stochastic matrix is given. How can I find a…
6
votes
3 answers
Duality with a stochastic matrix
If I have a stochastic matrix $X$- the sum of each row is equal to $1$ and all elements are non-negative.
Given this property, how can I show that:
$x'X=x'$ , $x\geq 0$
Has a non-zero solution?
I'm assuming this has something to do with proving a…
GBa
- 1,076
6
votes
0 answers
Second eigenvalue of a stochastic block matrix
Considering a stochastic block matrix in the form of,
$$\textbf{$P_{}$} = \begin{pmatrix} \textbf{$A_{}$} & \textbf{$B_{}$} \\ \textbf{$B_{}$} & \textbf{$A_{}$} \end{pmatrix}$$
I found out that the second largest eigenvalue $\lambda_{2}(P_{})$ of…
Udara
- 339