Frobenius normal formIn linear algebra, the Frobenius normal form or rational canonical form of a square matrix A with entries in a field F is a canonical form for matrices obtained by conjugation by invertible matrices over F. The form reflects a minimal decomposition of the vector space into subspaces that are cyclic for A (i.e., spanned by some vector and its repeated images under A). Since only one normal form can be reached from a given matrix (whence the "canonical"), a matrix B is similar to A if and only if it has the same rational canonical form as A.
PlatinumPlatinum is a chemical element with the symbol Pt and atomic number 78. It is a dense, malleable, ductile, highly unreactive, precious, silverish-white transition metal. Its name originates from Spanish platina, a diminutive of plata "silver". Platinum is a member of the platinum group of elements and group 10 of the periodic table of elements. It has six naturally occurring isotopes. It is one of the rarer elements in Earth's crust, with an average abundance of approximately 5 μg/kg.
Definite quadratic formIn mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every non-zero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite. A semidefinite (or semi-definite) quadratic form is defined in much the same way, except that "always positive" and "always negative" are replaced by "never negative" and "never positive", respectively.
PalladiumPalladium is a chemical element with the symbol Pd and atomic number 46. It is a rare and lustrous silvery-white metal discovered in 1803 by the English chemist William Hyde Wollaston. He named it after the asteroid Pallas, which was itself named after the epithet of the Greek goddess Athena, acquired by her when she slew Pallas. Palladium, platinum, rhodium, ruthenium, iridium and osmium form a group of elements referred to as the platinum group metals (PGMs).
Native metalA native metal is any metal that is found pure in its metallic form in nature. Metals that can be found as native deposits singly or in alloys include aluminium, antimony, arsenic, bismuth, cadmium, chromium, cobalt, indium, iron, manganese, molybdenum, nickel, niobium, rhenium, selenium, tantalum, tellurium, tin, titanium, tungsten, vanadium, and zinc, as well as the gold group (gold, copper, lead, aluminium, mercury, silver) and the platinum group (platinum, iridium, osmium, palladium, rhodium, ruthenium).
Spectrum (functional analysis)In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number is said to be in the spectrum of a bounded linear operator if either has no set-theoretic inverse; or the set-theoretic inverse is either unbounded or defined on a non-dense subset. Here, is the identity operator. By the closed graph theorem, is in the spectrum if and only if the bounded operator is non-bijective on .
Sulfide mineralThe sulfide minerals are a class of minerals containing sulfide (S2−) or disulfide (S22−) as the major anion. Some sulfide minerals are economically important as metal ores. The sulfide class also includes the selenides, the tellurides, the arsenides, the antimonides, the bismuthinides, the sulfarsenides and the sulfosalts. Sulfide minerals are inorganic compounds.
SpacetimeIn physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualizing and understanding relativistic effects such as how different observers perceive where and when events occur. Until the turn of the 20th century, the assumption had been that the three-dimensional geometry of the universe (its description in terms of locations, shapes, distances, and directions) was distinct from time (the measurement of when events occur within the universe).
TransposeIn linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix, often denoted by AT (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. In the case of a logical matrix representing a binary relation R, the transpose corresponds to the converse relation RT.
Diagonalizable matrixIn linear algebra, a square matrix is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix and a diagonal matrix such that , or equivalently . (Such , are not unique.) For a finite-dimensional vector space , a linear map is called diagonalizable if there exists an ordered basis of consisting of eigenvectors of .
Orthogonal matrixIn 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 where QT is the transpose of Q and I is the identity matrix. This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse: where Q−1 is the inverse of Q. An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary (Q−1 = Q∗), where Q∗ is the Hermitian adjoint (conjugate transpose) of Q, and therefore normal (Q∗Q = QQ∗) over the real numbers.
Three-dimensional spaceIn geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (coordinates) are required to determine the position of a point. Most commonly, it is the three-dimensional Euclidean space, the Euclidean n-space of dimension n=3 that models physical space. More general three-dimensional spaces are called 3-manifolds. Technically, a tuple of n numbers can be understood as the Cartesian coordinates of a location in a n-dimensional Euclidean space.
