# 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.

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Singular values of a rectangular stochastic matrix

I have a question related to rectangular stochastic matrix, ie for example a n x K matrix W such that the sum of the coefficients of a row is equal to one. Is there anything we can tell about the ...
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### 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 ...
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### 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 ...
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### 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 $x$ ...
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### 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 ...
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### 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 ...
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### 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 $R_k$ ...
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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 $$\tilde{Q}\,=\,(1-\alpha)... 2answers 1k views ### Convergence of iterated stochastic matrices It is well-known that for a stochastic aperiodic matrix M, the sequence (M^n)_n converges. Here I would like to a have a more precise analysis. Consider now a sequence of stochastic matrices (... 1answer 481 views ### The minimal norm of a shifted stochastic matrix Given a row-stochastic matrix M with singular values \sigma_{1} \geq \cdots \geq \sigma_{n}, I am looking for an upper bound on the expression$$\min_{\alpha} \left\| M- \frac{\alpha}{n}J_{n} \...
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?
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 ...