# 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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### 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 $\Sigma_{3\times3}$. ...
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### 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 ...
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### 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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### 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, ... 194 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 &...
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### 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?...
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### 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 ...
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### Apply doubly stochastic matrix M to a probability vector, then entropy increases?

Consider a vector $p =(p_1,...p_n)$, $p_i>0$, $\sum p_i = 1$ 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 ...
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### 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 ...
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### 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. ...
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### 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-...