Questions tagged [stochastic-matrices]
A stochastic matrix (also termed probability matrix, transition matrix, substitution matrix, or Markov matrix) is a square matrix used to describe the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability.
47 questions
0 votes
1 answer
82 views
Closure of the Karpelevič region
In 1938, Kolmogorov posed the problem of characterizing the set $$\Theta_n := \{ \lambda \in \mathbb{C} \mid \lambda \in \sigma(A),\ A \geqslant 0,\ Ae = e\},$$ where $e$ denotes the all-ones vector. ...
- 1,923
4 votes
1 answer
159 views
Fastest way to compute Cesàro limit of the powers of a stochastic matrix
Let $P$ be a (finite) stochastic matrix. Let $$ C = \lim_{n \to \infty} \frac{1}{n} \sum_{k=0}^{n-1} P^k $$ be the Cesàro limit of the powers of $P$. What is the fastest known way to compute $C$?
- 2,440
2 votes
3 answers
254 views
Quantifier-free description of the orthostochastic matrices
What is a quantifier-free formula defining the space of orthostochastic matrices? More precise version below. Fix $n \in {\Bbb N}$, let ${\mathcal M}_n$ denote the space of real $n\times n$ matrices, ...
- 688
6 votes
1 answer
359 views
Determinantal inequality for difference of substochastic matrices
Let $A=(A_{ij})_{1\le i,j\le n}$ be a square matrix with nonnegative real entries. Recall that $A$ is called a substochastic matrix if $$ \forall i,\ \ \sum_j A_{ij}\le 1\ . $$ In the course of my ...
- 23.6k
0 votes
0 answers
93 views
Random pseudo-inverse matrix problem
Given a matrix $M \in \mathbb{R}^{n \times N_d}$, $N_d \gg n$ and $\mathrm{rank}(M) = n$, the entries of $M$ are denoted as $M_{[ij]}, i = 1,...,n, j = 1,..., N_d$ and $M_{[ij]} \in [-\textbf{m}, \...
3 votes
1 answer
199 views
Comparison of time until absorption for two absorbing Markov chains
Let $\{X_t, t \geq 0\}$ and $\{X_t', t \geq 0\}$ denote two markov chains on the same state space $\{1, ..., n+1\}$ with transition probability matrices $P$ and $P'$ respectively. Suppose that both ...
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2 votes
0 answers
468 views
Ito lemma for SDEs on a Lie group
I'm trying to generalize the theorem described in this paper https://arxiv.org/abs/2001.01098 to the case of a semisimple compact matrix Lie group. In doing so i'm trying to define a formula ...
- 315
1 vote
1 answer
132 views
Compressing covariance matrix from 3X3 to 2X2
thanks for answering my question. My question is, Let $v_3=[a,b,c]^T$ is a probabilistic 3-D vector variable which is distributed by zero mean Gaussian of dense covariance matrix $\Sigma_{3\times3}$. ...
- 11
7 votes
2 answers
427 views
On permanent of a square of a doubly stochastic matrix
Let $A = (a_{i,j})$ be a double stochastic matrix with positive entries. That is, all entries are positive real numbers, and each row and column sums to one. A permanent of a matrix $A = (a_{i,j})$ is ...
user125543
-1 votes
2 answers
409 views
How to generate constant row and column sum matrices?
How can we randomly generate matrix $A \in \mathbb{R}^{n \times m}_{\geq 0}$ that satisfies $A 1_n = m1$ and $A^T 1_m = n1$.
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2 votes
0 answers
212 views
Is every nearly rank-1 doubly stochastic matrix a product of pairwise averaging matrices?
A doubly stochastic matrix is a square matrix with non-negative real entries where the sum of each row is $1$ and the sum of each column is $1$. A pairwise averaging matrix is a matrix of the form $tA+...
2 votes
0 answers
113 views
Maximum volume submatrices of a Khatri-Rao product of matrix exponentials
My question requires quite a bit of setup, which leads to a conjecture. So I split my question into three parts, Setup, Conjecture, and Question. Setup: Pick any two right stochastic matrices $\...
- 237
2 votes
0 answers
114 views
Pagerank Markov chain reductions
In short: if a Markov chain models a (generalized) pagerank, is it always possible to remove any of its state and obtain a Markov chain that models a pagerank close to the initial one? Full details. ...
- 1,534
1 vote
1 answer
472 views
Generalization of Sinkhorn’s theorem to stochastic tensors?
Is there an algorithm for mapping a nonnegative tensor / kD array into stochastic form? i.e. assume we have a tensor of unnormalized counts, $c: ℕ^{n^k}$, can we derive a function $f: ℕ^{n^k} → ℝ^{n^k}...
- 113
5 votes
2 answers
372 views
Existence of a specific stochastic matrix
Let $0\le x_1\le x_2\le \cdots\le x_n\le n-1$ be given. My question is as follows : Under which condition there exists a doubly stochastic matrix $M=(m_{i,j})_{1\le i,j\le n}$ s.t. $$\sum_{j=1}^n (j-1)...
user128095
1 vote
1 answer
579 views
The integral of a Gaussian process on a unit sphere
Suppose there exist a zero-mean Gaussian process $\mathbb{G} f_u$ indexed by $u \in \mathcal{S}^{p - 1}$ with known covariance $\mathrm{E} \big[ \mathbb{G} f_u \mathbb{G} f_v \big]$ when both $u$ and $...
- 341
1 vote
1 answer
728 views
Follow up: Show that these vectors are linearly independent almost surely
I posted this question some time ago here. I started a bounty for it and received an answer which helped me a lot. However, I still have some issues I want to discuss regarding it. Unfortunately I can'...
- 354
11 votes
1 answer
1k views
Show that these vectors are linearly independent almost surely
So I'm doing research in control theory and I have been stuck with this problem for a while. Let me explain my issue, then my proposal, and finally my concrete question. Problem: I have $m<n$ real $...
- 354
5 votes
2 answers
406 views
Doubly-stochastic partial-isometric matrices
An $n\times n$ matrix $A$ with nonegative real entries $a_{ij}$ is said to be doubly stochastic if $\sum_{i=1}^na_{ij} = 1$, for all $j$, and $\sum_{j=1}^na_{ij}=1$, for all $i$. Much is known [1] ...
- 2,313
1 vote
0 answers
420 views
Upper and lower bounds on the entries of a matrix power
Say I have a non-negative square $n\times n$ irreducible stochastic matrix $A$ (i.e., each column sums to 1), for which the following holds: $$A_{ij} > 0 \iff A_{ji} > 0.$$ I know that no more ...
- 161
2 votes
1 answer
231 views
Diagonalizable stochastic matrix that satisfies an equation
Given an arbitrary discrete probability distribution $a = (a_1, ..., a_n)$ and another arbitrary discrete probability distribution $b = (b_1, ..., b_n)$, what is the easiest known way to find a ...
- 23
3 votes
1 answer
166 views
Is the Wasserstein kernel positive definite?
Define a point cloud $X=\{x_i\}_{1\leq i\leq n}$, for $x_i\in\mathbb R^d$. Define the Wasserstein kernel as $$W(X,Y)=\max_{T}\frac{1}{n}\sum_{kl}T_{kl}\langle x_k,y_l \rangle$$ where $T$ is any doubly ...
- 345
2 votes
0 answers
81 views
Completeness results for Categorical Quantum Mechanics restricted to one $\dagger$-Frobenius Algebra?
I've seen the various completeness results for Categorical Quantum Mechanics (CQM) axiom systems involving two interacting Frobenius algebras with various restrictions of phase-nodes. For example, the ...
- 21
20 votes
1 answer
2k views
Is Van der Waerden's conjecture really due to Van der Waerden?
Van der Waerden's conjecture (now a theorem of Egorychev and Falikman) states that the permanent of a doubly stochastic matrix is at least $n!/n^n$. The Wikipedia article, as well as many other ...
- 87.7k
4 votes
2 answers
545 views
Frobenius normal form of a doubly stochastic matrix
If $A \in M_n(\mathbb{C})$, then $A$ is called reducible if there is a permuation matrix $P$ such that $$ P^\top A P = \begin{bmatrix} A_{11} & A_{12} \\ 0 & A_{22} \end{bmatrix}, $$ in ...
- 1,923
8 votes
0 answers
455 views
Eigenvalues of cyclic stochastic matrices
Let's consider the following $n \times n$ cyclic stochastic matrix $$ M= \begin{pmatrix} 0 & a_2 & & & &b_n \\\ b_1 & 0& a_3& &&& \\\ & b_2 & ...
- 181
11 votes
0 answers
773 views
What properties characterize the function $L(x) = x+\exp(x) \log(x)$?
As might be known, the function $L(x) = x+\exp(x)\log(x)$ plays a prominent role in the Lagarias formulation of the Riemann hypothesis: $$\sigma(n) \le H_n + \exp(H_n) \log(H_n)$$ My question is, ...
user6671
3 votes
1 answer
272 views
Mixing time and spectral gap for a special stochastic matrix
Consider the following dimension stochastic matrix, \begin{bmatrix} p & q & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 &...
- 103
0 votes
1 answer
170 views
Pre-garbling does not improve capacity of a channel
Suppose ABC=B for some column stochastic matrices A, B, and C. Can the following implication be made without further restrictions: There necessarily exists a column stochastic matrix D such that DB=BC?...
- 9
4 votes
0 answers
319 views
Birkhoff – von Neumann for "$k$-stochastic matrices"
Recall that a doubly-stochastic matrix is a square matrix with non-negative elements such the sum of the elements in every row, as well as in every column, is $1$. The set of doubly-stochastic ...
- 23.4k
3 votes
4 answers
2k views
Apply doubly stochastic matrix M to a probability vector, then entropy increases?
Consider a vector $p =(p_1,\dots,p_n)$, $p_i>0$, $\sum p_i = 1$ and a matrix $M_{ij}$, which is doubly stochastic: $\sum_i M_{ij} = 1, \sum_j M_{ij} = 1, M_{ij} > 0$. Question 1 Just apply ...
- 26.3k
1 vote
1 answer
675 views
On the eigenvalue of the expectation value of a random matrix in quadratic form
When we handle with some dynamic input-output mappings, there occurs a question as follows: Let $M$ be a random matrix, of which each element contains random terms. Consider the two expectation ...
- 77
1 vote
0 answers
477 views
Sufficient conditions for all eigenvalues simple in stochastic matrix
The "largest" eigenvalue $1$ of a stochastic matrix is well-characterized by the classical Perron-Frobenius theorem. In particular, it gives sufficient conditions for the eigenvalue $1$ to be simple. ...
- 11
3 votes
1 answer
136 views
Bounds on $\|(P+\Delta)^n - P^n\|_F$ for stochastic matrices
Let us suppose that $P$ is a stochastic matrix (non-negative matrix with $P \mathbf{1} = \mathbf{1}$). Let $\Delta$ such that $P + \Delta$ is a stochastic matrix (which means $P + \Delta$ is non-...
- 113
8 votes
1 answer
435 views
On the limit of partial sum of infinite doubly stochastic matrix
Let $A=(a_{ij})$ be an infinite doubly stochastic matrix. Does there necessarily exist a subsequence $\{n_k\}_{k=1}^\infty$ such that $$ \lim_{k\to\infty}\frac{1}{n_k}\sum_{i=1}^{n_k}\sum_{j=1}^{n_k}...
- 171
2 votes
1 answer
234 views
A question on the partial sum of infinite doubly stochastic matrix
Let $A=(a_{ij})$ be an infinite doubly stochastic matrix. Is the following statement true ? $$ \lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n\sum_{j=1}^na_{ij} >0 $$ Any reference or comment on this is ...
- 171
2 votes
0 answers
438 views
Constraint involving a stochastic matrix and its inverse
Consider the following feasibility problem: Find an $n \times n$ stochastic matrix $L$ such that $L^{-1} M L x$ is a non-negative vector, where $M$ is a known $n \times n$ positive matrix and $x$ is ...
- 121
13 votes
2 answers
968 views
The expected square of the determinant of a random row stochastic matrix
In this question Anthony Quas asks about the expected absolute value of the determinant of an $n\times n$ row stochastic matrix $A$, where the rows are independently selected from the uniform ...
- 53.5k
4 votes
1 answer
857 views
Determinant of a random row stochastic matrix
Does anyone know anything about the determinant of a random $n\times n$ row stochastic matrix? What I have in mind is that the rows are independently selected from the uniform distribution on the unit ...
- 23.6k
2 votes
4 answers
352 views
Find a square, stochastic matrix of odd size, not a permutation matrix, with an eigenvalue other than 1 on the unit circle
...or prove that none exists. Note that such a matrix $M$ couldn't be primitive, so there would be at least one entry equal to zero in every power $M^k$ (Perron-Frobenius theory). Preferably the ...
- 121
4 votes
0 answers
507 views
Spectral radius of the product of a right stochastic matrix and a block diagonal matrix
Let us define the following matrix: $C=AB$ where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$ … $B_N$, each of dimensions $M \times M$. I know that $B_k = I_M - \mu R_k$ with $R_k$ ...
- 41
3 votes
1 answer
409 views
Eigenvectors of a perturbed reducible stochastic matrix
Let $Q$ be a $n\times n$ reducible stochastic matrix. Let $J$ be such that $[J]_{ij}={1 \over n}$. Now for a small positive constant $\alpha\in [0,1]$, consider the matrix $$\tilde{Q}\,=\,(1-\alpha)...
- 1,421
9 votes
1 answer
465 views
Bound on the ratio of top 2 eigenvalues
Let $P$ be a $n \times n$ stochastic matrix such that $P_{ij}=\tau$ if $i \neq j$ and $P_{ii} = 1 - (n-1)\tau$ where $0<\tau < \frac{1}{n}$. It is clear that the largest eigenvalue of $P$ is $1$,...
user31317
3 votes
2 answers
2k views
Convergence of iterated stochastic matrices
It is well-known that for a stochastic aperiodic matrix $M$, the sequence $(M^n)_n$ converges. Here I would like to a have a more precise analysis. Consider now a sequence of stochastic matrices $(...
- 31
5 votes
1 answer
739 views
The minimal norm of a shifted stochastic matrix
Given a row-stochastic matrix $M$ with singular values $\sigma_{1} \geq \cdots \geq \sigma_{n}$, I am looking for an upper bound on the expression $$\min_{\alpha} \left\| M- \frac{\alpha}{n}J_{n} \...
- 235
15 votes
3 answers
4k views
Non-diagonalizable doubly stochastic matrices
Are there constructive examples of doubly stochastic matrices (whose rows and columns all sum up to $1$ and contain only non-negative entries) that are not diagonalizable?
- 731
22 votes
3 answers
1k views
Which doubly stochastic matrices can be written as products of pairwise averaging matrices?
A matrix $A$ is called doubly stochastic if its entries are nonnegative, and if all of its rows and columns add up to $1$. A subset of doubly stochastic matrices is the set of pairwise averaging ...
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