Projective unitary groupIn mathematics, the projective unitary group PU(n) is the quotient of the unitary group U(n) by the right multiplication of its center, U(1), embedded as scalars. Abstractly, it is the holomorphic isometry group of complex projective space, just as the projective orthogonal group is the isometry group of real projective space. In terms of matrices, elements of U(n) are complex n×n unitary matrices, and elements of the center are diagonal matrices equal to eiθ multiplied by the identity matrix.
HyperplaneIn geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. This notion can be used in any general space in which the concept of the dimension of a subspace is defined. In different settings, hyperplanes may have different properties.
Eigenvalues and eigenvectorsIn linear algebra, an eigenvector (ˈaɪgənˌvɛktər) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a constant factor when that linear transformation is applied to it. The corresponding eigenvalue, often represented by , is the multiplying factor. Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The eigenvectors for a linear transformation matrix are the set of vectors that are only stretched, with no rotation or shear.
Special unitary groupIn mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 in the special case. The group operation is matrix multiplication. The special unitary group is a normal subgroup of the unitary group U(n), consisting of all n×n unitary matrices. As a compact classical group, U(n) is the group that preserves the standard inner product on .
Eigendecomposition of a matrixIn linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition", derived from the spectral theorem. Eigenvalue, eigenvector and eigenspace A (nonzero) vector v of dimension N is an eigenvector of a square N × N matrix A if it satisfies a linear equation of the form for some scalar λ.
United StatesThe United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America and consisting of 50 states, a federal district, five major unincorporated territories, nine Minor Outlying Islands, and 326 Indian reservations. It is the world's third-largest country by both land and total area. It shares land borders with Canada to its north and with Mexico to its south and has maritime borders with the Bahamas, Cuba, Russia, and other nations.
Chinese classifierThe modern Chinese varieties make frequent use of what are called classifiers or measure words. One use of classifiers is when a noun is qualified by a numeral or demonstrative. In the Chinese equivalent of a phrase such as "three books" or "that person", it is normally necessary to insert an appropriate classifier between the numeral/demonstrative and the noun. For example, in Standard Mandarin, the first of these phrases would be 三书 sān shū, where sān means "three", shū means "books", and běn is the required classifier.
IsraelIsrael (ˈɪzri.əl,_-reɪ-; יִשְׂרָאֵל Yīsrāʾēl jisʁaˈʔel; إِسْرَائِيل ʾIsrāʾīl), officially the State of Israel (מְדִינַת יִשְׂרָאֵל Medīnat Yīsrāʾēl mediˈnat jisʁaˈʔel; دَوْلَة إِسْرَائِيل Dawlat Isrāʾīl), is a country in West Asia. It is bordered by Lebanon to the north, by Syria to the northeast, by Jordan to the east, by the Red Sea to the south, by Egypt to the southwest, by the Mediterranean Sea to the west, and by the Palestinian territories - the West Bank along the east and the Gaza Strip along the southwest.
QR decompositionIn linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix A into a product A = QR of an orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem and is the basis for a particular eigenvalue algorithm, the QR algorithm. Any real square matrix A may be decomposed as where Q is an orthogonal matrix (its columns are orthogonal unit vectors meaning ) and R is an upper triangular matrix (also called right triangular matrix).
Classifier (linguistics)A classifier (abbreviated or ) is a word or affix that accompanies nouns and can be considered to "classify" a noun depending on the type of its referent. It is also sometimes called a measure word or counter word. Classifiers play an important role in certain languages, especially East Asian languages, including Chinese, Korean, Japanese, and Vietnamese. Classifiers are absent or marginal in European languages. An example of a possible classifier in English is piece in phrases like "three pieces of paper".
Unit of measurementA unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multiple of the unit of measurement. For example, a length is a physical quantity. The metre (symbol m) is a unit of length that represents a definite predetermined length. For instance, when referencing "10 metres" (or 10 m), what is actually meant is 10 times the definite predetermined length called "metre".
United NationsThe United Nations, referred to informally as the UN, is an intergovernmental organization whose stated purposes are to maintain international peace and security, develop friendly relations among nations, achieve international cooperation, and serve as a centre for harmonizing the actions of nations. It is the world's largest international organization. The UN is headquartered in New York City (in the United States, but with certain extraterritorial privileges), and the organization has other offices in Geneva, Nairobi, Vienna, and The Hague, where the International Court of Justice is headquartered.
