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A doubly stochastic matrix is a square matrix of nonnegative real numbers with each row and column summing to 1. A substochastic matrix is a real square matrix whose row sums are all ; In the same vein, one may define a probability vector as a vector whose elements are nonnegative real numbers which sum to 1. Thus, each row of a right ...
In nuclear physics, random matrices were introduced by Eugene Wigner to model the nuclei of heavy atoms. [1] [2] Wigner postulated that the spacings between the lines in the spectrum of a heavy atom nucleus should resemble the spacings between the eigenvalues of a random matrix, and should depend only on the symmetry class of the underlying evolution. [4]
Stochastic (/ s t ə ˈ k æ s t ɪ k /; from Ancient Greek στόχος (stókhos) 'aim, guess') [1] is the property of being well-described by a random probability distribution. [1] Stochasticity and randomness are technically distinct concepts: the former refers to a modeling approach, while the latter describes phenomena; in everyday ...
The class of doubly stochastic matrices is a convex polytope known as the Birkhoff polytope.Using the matrix entries as Cartesian coordinates, it lies in an ()-dimensional affine subspace of -dimensional Euclidean space defined by independent linear constraints specifying that the row and column sums all equal 1.
A Markov chain can be described by a stochastic matrix, which lists the probabilities of moving to each state from any individual state. From this matrix, the probability of being in a particular state n steps in the future can be calculated. A Markov chain's state space can be partitioned into communicating classes that describe which states ...
In mathematics and statistics, a probability vector or stochastic vector is a vector with non-negative entries that add up to one.. The positions (indices) of a probability vector represent the possible outcomes of a discrete random variable, and the vector gives us the probability mass function of that random variable, which is the standard way of characterizing a discrete probability ...
The correlation matrix (also called second moment) of an random vector is an matrix whose (i,j) th element is the correlation between the i th and the j th random variables. The correlation matrix is the expected value, element by element, of the matrix computed as , where the superscript T refers to the transpose of the indicated vector: [4 ...
The random walk normalized Laplacian is defined as := + = + where D is the degree matrix. Since the degree matrix D is diagonal, its inverse + is simply defined as a diagonal matrix, having diagonal entries which are the reciprocals of the corresponding diagonal entries of D.