LU decompositionIn numerical analysis and linear algebra, lower–upper (LU) decomposition or factorization factors a matrix as the product of a lower triangular matrix and an upper triangular matrix (see matrix decomposition). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix.
DiboraneDiborane(6), commonly known as diborane, is the chemical compound with the formula B2H6. It is a toxic, colorless, and pyrophoric gas with a repulsively sweet odor. Given its simple formula, borane is a fundamental boron compound. It has attracted wide attention for its electronic structure. Several of its derivatives are useful reagents. The structure of diborane has D2h symmetry. Four hydrides are terminal, while two bridge between the boron centers. The lengths of the B–Hbridge bonds and the B–Hterminal bonds are 1.
Cholesky decompositionIn linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ʃəˈlɛski ) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations. It was discovered by André-Louis Cholesky for real matrices, and posthumously published in 1924. When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations.
BoraneBorane, also known as borine, is an unstable and highly reactive molecule with the chemical formula BH3. The preparation of borane carbonyl, BH3(CO), played an important role in exploring the chemistry of boranes, as it indicated the likely existence of the borane molecule. However, the molecular species BH3 is a very strong Lewis acid. Consequently, it is highly reactive and can only be observed directly as a continuously produced, transitory, product in a flow system or from the reaction of laser ablated atomic boron with hydrogen.
Chemical decompositionChemical decomposition, or chemical breakdown, is the process or effect of simplifying a single chemical entity (normal molecule, reaction intermediate, etc.) into two or more fragments. Chemical decomposition is usually regarded and defined as the exact opposite of chemical synthesis. In short, the chemical reaction in which two or more products are formed from a single reactant is called a decomposition reaction. The details of a decomposition process are not always well defined.
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).
Gas chromatographyGas chromatography (GC) is a common type of chromatography used in analytical chemistry for separating and analyzing compounds that can be vaporized without decomposition. Typical uses of GC include testing the purity of a particular substance, or separating the different components of a mixture. In preparative chromatography, GC can be used to prepare pure compounds from a mixture. Gas chromatography is also sometimes known as vapor-phase chromatography (VPC), or gas–liquid partition chromatography (GLPC).
Polar decompositionIn mathematics, the polar decomposition of a square real or complex matrix is a factorization of the form , where is a unitary matrix and is a positive semi-definite Hermitian matrix ( is an orthogonal matrix and is a positive semi-definite symmetric matrix in the real case), both square and of the same size. Intuitively, if a real matrix is interpreted as a linear transformation of -dimensional space , the polar decomposition separates it into a rotation or reflection of , and a scaling of the space along a set of orthogonal axes.
HydrideIn chemistry, a hydride is formally the anion of hydrogen (H−), a hydrogen atom with two electrons. The term is applied loosely. At one extreme, all compounds containing covalently bound H atoms are called hydrides: water (H2O) is a hydride of oxygen, ammonia is a hydride of nitrogen, etc. For inorganic chemists, hydrides refer to compounds and ions in which hydrogen is covalently attached to a less electronegative element. In such cases, the H centre has nucleophilic character, which contrasts with the protic character of acids.
BoranesA borane is a compound with the formula BxHy or a related anion. Many such boranes are known. Most common are those with 1 to 12 boron atoms. Although they have few practical applications, the boranes exhibit structures and bonding that differs strongly from the patterns seen in hydrocarbons. Hybrids of boranes and hydrocarbons, the carboranes are also well developed. The development of the chemistry of boranes led to innovations in synthetic methods as well as structure and bonding.
