สอบถามเกี่ยวกับงานวิจัย เรื่อง symmetric matrices and their doubly stochastic shapes
 

เรื่อง Sinkhorn limits of 3x3 symmetric matrices and their doubly stochastic shapes

Let A and B be positive nxn matrices. We write $A \sim B$ if there exist nxn
permutation matrices P and Q and $\lambda > 0$ such that
$B =\lambda PAQ$
This is an equivalence relation. Moreover, $A \sim B$ implies
$S(B) = \lambda PS(A)Q$
Thus, it suffices to determine the Sinkhorn limit of only one matrix in an equivalence class.
We shall compute the Sinkhorn limit of every symmetric positive 3x3 matrix
whose set of coordinates consists of two distinct real numbers.
Let A be such a matrix with coordinates M and N with M = N. There are 9
coordinate positions in the matrix, and so exactly one of the numbers M and N
occurs at least five times. Suppose that the coordinate M occurs five or more times.
Let $\lambda = 1/M$ and $K = N/M$ The matrix $\lambda A$ has two distinct positive coordinates
1 and K, and K occurs at most four times. There are seven equivalence classes of
such matrices with respect to permutations and dilations. The main result of this
paper is the calculation of the Sinkhorn limits of these matrices.

ผมสงสัยว่าทำไมถึงมี 7 และก็หามายังไงครับ ขอบคุณมากครับ
อันนี้ลิ้งเปเปอร์ครับ https://arxiv.org/pdf/1905.09426.pdf

In mathematics, especially in probability and combinatorics, a doubly stochastic matrix
(also called bistochastic matrix), is a square matrix of nonnegative real numbers,
each of whose rows and columns sums to 1,[1]

Thus, a doubly stochastic matrix is both left stochastic and right stochastic.[1][2]

Indeed, any matrix that is both left and right stochastic must be square:
if every row sums to one then the sum of all entries in the matrix must be equal to the number of rows,
and since the same holds for columns, the number of rows and columns must be equal.[1]

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.