What is doubly stochastic matrix?

A doubly stochastic matrix is a square matrix in which both the rows and columns add up to 1.

In other words, a matrix A = [aij] is said to be doubly stochastic if:

1. Σ aij = 1 for all i
2. Σ aij = 1 for all j

Since the rows and columns of a doubly stochastic matrix sum to 1, each entry in the matrix represents the probability of transitioning from one state to another in a Markov chain.

Doubly stochastic matrices have a number of interesting properties and applications in various fields including probability theory, optimization, and economics. They are also closely related to the concept of doubly stochastic processes.

In addition, a doubly stochastic matrix can be considered as a permutation matrix multiplied by a diagonal matrix with all positive entries. This relationship is useful in analyzing the properties of doubly stochastic matrices and their manipulation in various mathematical operations.