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Unistochastic
In mathematics, a unistochastic matrix (also called ''unitary-stochastic'') is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some unitary matrix. A square matrix ''B'' of size ''n'' is doubly stochastic (or ''bistochastic'') if all its entries are non-negative real numbers and each of its rows and columns sum to 1. It is unistochastic if there exists a unitary matrix ''U'' such that : B_=, U_, ^2 \text i,j=1,\dots,n. \, This definition is analogous to that for an orthostochastic matrix, which is a doubly stochastic matrix whose entries are the squares of the entries in some orthogonal matrix. Since all orthogonal matrices are necessarily unitary matrices, all orthostochastic matrices are also unistochastic. The converse, however, is not true. First, all 2-by-2 doubly stochastic matrices are both unistochastic and orthostochastic, but for larger ''n'' this is not the case. For example, take n=3 and consider the following doubl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthostochastic Matrix
In mathematics, an orthostochastic matrix is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some orthogonal matrix. The detailed definition is as follows. A square matrix ''B'' of size ''n'' is doubly stochastic (or ''bistochastic'') if all its rows and columns sum to 1 and all its entries are nonnegative real numbers. It is orthostochastic if there exists an orthogonal matrix ''O'' such that : B_=O_^2 \text i,j=1,\dots,n. \, All 2-by-2 doubly stochastic matrices are orthostochastic (and also unistochastic In mathematics, a unistochastic matrix (also called ''unitary-stochastic'') is a doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some unitary matrix. A square matrix ''B'' of size ''n'' is doubly stoc ...) since for any : B= \begin a & 1-a \\ 1-a & a \end we find the corresponding orthogonal matrix : O = \begin \cos \phi & \sin \phi \\ - \sin \phi & \cos \phi \end, with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doubly Stochastic Matrix
In mathematics, especially in probability and combinatorics, a doubly stochastic matrix (also called bistochastic matrix) is a square matrix X=(x_) of nonnegative real numbers, each of whose rows and columns sums to 1, i.e., :\sum_i x_=\sum_j x_=1, Thus, a doubly stochastic matrix is both left stochastic and right stochastic. Indeed, any matrix that is both left and right stochastic must be square: if every row sums to 1 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. Birkhoff polytope The class of n\times n doubly stochastic matrices is a convex polytope known as the Birkhoff polytope B_n. Using the matrix entries as Cartesian coordinates, it lies in an (n-1)^2-dimensional affine subspace of n^2-dimensional Euclidean space defined by 2n-1 independent linear constraints specifying that the row and column sums all equal 1. (There are 2n-1 constraints rather than 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unitary Matrix
In linear algebra, an invertible complex square matrix is unitary if its matrix inverse equals its conjugate transpose , that is, if U^* U = UU^* = I, where is the identity matrix. In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (), so the equation above is written U^\dagger U = UU^\dagger = I. A complex matrix is special unitary if it is unitary and its matrix determinant equals . For real numbers, the analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes. Properties For any unitary matrix of finite size, the following hold: * Given two complex vectors and , multiplication by preserves their inner product; that is, . * is normal (U^* U = UU^*). * is diagonalizable; that is, is unitarily similar to a diagonal matrix, as a consequence of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity matrix. This leads to the equivalent characterization: a matrix is orthogonal if its transpose is equal to its inverse: Q^\mathrm=Q^, where is the inverse of . An orthogonal matrix is necessarily invertible (with inverse ), unitary (), where is the Hermitian adjoint ( conjugate transpose) of , and therefore normal () over the real numbers. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the inner product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation, reflection or rotoreflection. In other words, it is a unitary transformation. The set of orthogonal matrices, under multiplication, forms the group , known as th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proper Subset
In mathematics, a set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. When quantified, A \subseteq B is represented as \forall x \left(x \in A \Rightarrow x \in B\right). One can prove the statement A \subseteq B by applying a proof technique known as the element argument:Let sets ''A'' and ''B'' be given. To prove that A \subseteq B, # suppose that ''a'' is a particular but arbitrarily chosen element of A # show that ''a'' is an element of ''B''. The validity of this technique can be seen as a consequence of univers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation Matrix
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column with all other entries 0. An permutation matrix can represent a permutation of elements. Pre- multiplying an -row matrix by a permutation matrix , forming , results in permuting the rows of , while post-multiplying an -column matrix , forming , permutes the columns of . Every permutation matrix ''P'' is orthogonal, with its inverse equal to its transpose: P^=P^\mathsf. Indeed, permutation matrices can be characterized as the orthogonal matrices whose entries are all non-negative. The two permutation/matrix correspondences There are two natural one-to-one correspondences between permutations and permutation matrices, one of which works along the rows of the matrix, the other along its columns. Here is an example, starting with a permutation in two-line form at the upper left: :\begin \pi\colon\begin1&2&3&4\\3&2&4&1\e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a Bounded set, bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its projective duality, dual problem of intersecting Half-space (geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhoff Polytope
The Birkhoff polytope ''B''''n'' (also called the assignment polytope, the polytope of doubly stochastic matrices, or the perfect matching polytope of the complete bipartite graph K_) is the convex polytope in R''N'' (where ''N'' = ''n''2) whose points are the doubly stochastic matrices, i.e., the matrices whose entries are non-negative real numbers and whose rows and columns each add up to 1. It is named after Garrett Birkhoff. Properties Vertices The Birkhoff polytope has ''n''! vertices, one for each permutation on ''n'' items. This follows from the Birkhoff–von Neumann theorem, which states that the extreme points of the Birkhoff polytope are the permutation matrices, and therefore that any doubly stochastic matrix may be represented as a convex combination of permutation matrices; this was stated in a 1946 paper by Garrett Birkhoff, but equivalent results in the languages of projective configurations and of regular bipartite graph matchings, respectively, were sho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jarlskog Invariant
Jarlskog is a Swedish surname, where 'skog' means a forest. Notable people with the surname include: * Cecilia Jarlskog (born 1941), Swedish theoretical physicist * Ida Jarlskog (born 1998), Swedish tennis player {{surname Swedish-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
