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Eigenvalues and eigenvectors

Index Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it. [1]

235 relations: Abel–Ruffini theorem, Accuracy and precision, Addison-Wesley, Adjacency matrix, Alfred Clebsch, Algebra representation, Algebraic number, Algebraic solution, Angular frequency, Antieigenvalue theory, Arthur Cayley, Associative algebra, Atomic orbital, Atomic physics, Augustin-Louis Cauchy, Banach space, Basis (linear algebra), Basis set (chemistry), Bioinformatics, Biometrics, Bound state, Bounded operator, Bra–ket notation, Brady Haran, Brightness, Center of mass, Characteristic polynomial, Charles Hermite, Chemometrics, Clastic rock, Closure (mathematics), Coefficient, Collinearity, Combinatorial explosion, Commutative property, Companion matrix, Compass rose, Complex conjugate, Complex number, Conjugate transpose, Correlation and dependence, Covariance matrix, Data compression, Data mining, Data set, David Hilbert, Deeplearning4j, Defective matrix, Degree of a polynomial, Degrees of freedom (mechanics), ..., Derivative, Determinant, Diagonal, Diagonal matrix, Diagonalizable matrix, Differential equation, Differential operator, Digital image processing, Direct sum, Discrete Laplace operator, Discriminant, Distributive property, Dot product, Earth Surface Processes and Landforms, Eigendecomposition of a matrix, Eigenface, Eigenfunction, Eigenplane, Eigenvalue algorithm, Eigenvector centrality, Energy, Explained variation, Exponential function, Facial recognition system, Factor analysis, Factorization, Field (mathematics), Finite element method, Floating-point arithmetic, Fock matrix, Formal power series, Francesco Brioschi, Function space, Functional analysis, Fundamental theorem of algebra, Funk & Wagnalls, Gaussian elimination, Generalized eigenvector, Geology, German language, Google, Graph theory, Hamiltonian (quantum mechanics), Hartree–Fock method, Heat equation, Henk van der Vorst, Henri Poincaré, Hermann Schwarz, Hermann von Helmholtz, Hermitian matrix, Hilbert space, Houghton Mifflin Harcourt, Householder transformation, Identity matrix, Inertia, Integral transform, Intermediate value theorem, Introduction to eigenstates, Invariant subspace, Inverse iteration, Invertible matrix, Ionization energy, Irrational number, Iteration, Iterative method, Jacques Charles François Sturm, James Demmel, Johann Andreas Segner, John G. F. Francis, John Wiley & Sons, Jordan normal form, Joseph Fourier, Joseph Liouville, Joseph-Louis Lagrange, Karl Weierstrass, Kernel (linear algebra), Koopmans' theorem, Lanczos algorithm, Laplace's equation, Laplacian matrix, Leibniz formula for determinants, Leonhard Euler, Linear algebra, Linear combination, Linear equation, Linear map, Linear subspace, Linear system, List of numerical analysis software, Marketing, Markov chain, Mass matrix, MathWorld, Matrix (mathematics), Matrix decomposition, Matrix multiplication, Matrix similarity, Mechanics, Module (mathematics), Molecular orbital, Molecular physics, Moment of inertia, Mona Lisa, Multiplicity (mathematics), Multivariate statistics, Nonlinear eigenproblem, Numerical method, Observable, Orthogonal basis, Orthogonal matrix, Orthogonality, PageRank, Parallel (geometry), Permutation matrix, Pixel, Poisson's equation, Polynomial, Polynomial long division, Positive-definite matrix, Power iteration, Principal component analysis, Psychometrics, Q methodology, QR algorithm, Quadratic eigenvalue problem, Quadratic equation, Quadratic form, Quadric, Quantum chemistry, Quantum mechanics, Quantum state, Rational number, Reciprocal polynomial, Recurrence relation, Representation theory, Richard von Mises, Rigid body, Roothaan equations, Rotation (mathematics), Round-off error, Row and column vectors, Scalar (mathematics), Scalar multiplication, Scaling (geometry), Schrödinger equation, Self-adjoint operator, Separation of variables, Set (mathematics), Shear mapping, Singular value, Skew-symmetric matrix, Solid mechanics, Sparse matrix, Spectral clustering, Spectral graph theory, Spectrum (functional analysis), Square matrix, Square-integrable function, Squeeze mapping, Stability theory, Stationary distribution, Statistical hypothesis testing, Statistical significance, Stiffness matrix, Stress (mechanics), Structural equation modeling, Sturm–Liouville theory, Symmetric matrix, Tensor, Till, Trace (linear algebra), Triangular matrix, Turn (geometry), Union (set theory), Unit circle, Unitary matrix, University of Nottingham, Variance, Vector space, Vera Kublanovskaya, Vibration, Wave function, Weight (representation theory), Wilkinson's polynomial, Zero element. Expand index (185 more) »

Abel–Ruffini theorem

In algebra, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no algebraic solution—that is, solution in radicals—to the general polynomial equations of degree five or higher with arbitrary coefficients.

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Accuracy and precision

Precision is a description of random errors, a measure of statistical variability.

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Addison-Wesley

Addison-Wesley is a publisher of textbooks and computer literature.

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Adjacency matrix

In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph.

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Alfred Clebsch

Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory.

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Algebra representation

In abstract algebra, a representation of an associative algebra is a module for that algebra.

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Algebraic number

An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).

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Algebraic solution

An algebraic solution or solution in radicals is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of an algebraic equation in terms of the coefficients, relying only on addition, subtraction, multiplication, division, raising to integer powers, and the extraction of nth roots (square roots, cube roots, and other integer roots).

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Angular frequency

In physics, angular frequency ω (also referred to by the terms angular speed, radial frequency, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate.

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Antieigenvalue theory

In applied mathematics, antieigenvalue theory was developed by Karl Gustafson from 1966 to 1968.

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Arthur Cayley

Arthur Cayley F.R.S. (16 August 1821 – 26 January 1895) was a British mathematician.

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Associative algebra

In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field.

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Atomic orbital

In quantum mechanics, an atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom.

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Atomic physics

Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus.

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Augustin-Louis Cauchy

Baron Augustin-Louis Cauchy FRS FRSE (21 August 178923 May 1857) was a French mathematician, engineer and physicist who made pioneering contributions to several branches of mathematics, including: mathematical analysis and continuum mechanics.

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Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

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Basis (linear algebra)

In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.

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Basis set (chemistry)

A basis set in theoretical and computational chemistry is a set of functions (called basis functions) that is used to represent the electronic wave function in the Hartree–Fock method or density-functional theory in order to turn the partial differential equations of the model into algebraic equations suitable for efficient implementation on a computer.

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Bioinformatics

Bioinformatics is an interdisciplinary field that develops methods and software tools for understanding biological data.

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Biometrics

Biometrics is the technical term for body measurements and calculations.

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Bound state

In quantum physics, a bound state is a special quantum state of a particle subject to a potential such that the particle has a tendency to remain localised in one or more regions of space.

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Bounded operator

In functional analysis, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded above by the same number, over all non-zero vectors v in X. In other words, there exists some M\ge 0 such that for all v in X The smallest such M is called the operator norm \|L\|_ \, of L. A bounded linear operator is generally not a bounded function; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless L(v).

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Bra–ket notation

In quantum mechanics, bra–ket notation is a standard notation for describing quantum states.

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Brady Haran

Brady John Haran (born 18 June 1976) is an Australian-born British independent filmmaker and video journalist who is known for his educational videos and documentary films produced for BBC News and his YouTube channels, the most notable being Periodic Videos and Numberphile.

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Brightness

Brightness is an attribute of visual perception in which a source appears to be radiating or reflecting light.

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Center of mass

In physics, the center of mass of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero, or the point where if a force is applied it moves in the direction of the force without rotating.

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Characteristic polynomial

In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots.

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Charles Hermite

Prof Charles Hermite FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.

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Chemometrics

Chemometrics is the science of extracting information from chemical systems by data-driven means.

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Clastic rock

Clastic rocks are composed of fragments, or clasts, of pre-existing minerals and rock.

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Closure (mathematics)

A set has closure under an operation if performance of that operation on members of the set always produces a member of the same set; in this case we also say that the set is closed under the operation.

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Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but may be any expression.

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Collinearity

In geometry, collinearity of a set of points is the property of their lying on a single line.

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Combinatorial explosion

In mathematics, a combinatorial explosion is the rapid growth of the complexity of a problem due to how the combinatorics of the problem is affected by the input, constraints, and bounds of the problem.

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Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

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Companion matrix

In linear algebra, the Frobenius companion matrix of the monic polynomial p(t).

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Compass rose

A compass rose, sometimes called a windrose or Rose of the Winds, is a figure on a compass, map, nautical chart, or monument used to display the orientation of the cardinal directions (north, east, south, and west) and their intermediate points.

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Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.

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Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

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Conjugate transpose

In mathematics, the conjugate transpose or Hermitian transpose of an m-by-n matrix A with complex entries is the n-by-m matrix A∗ obtained from A by taking the transpose and then taking the complex conjugate of each entry.

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Correlation and dependence

In statistics, dependence or association is any statistical relationship, whether causal or not, between two random variables or bivariate data.

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Covariance matrix

In probability theory and statistics, a covariance matrix (also known as dispersion matrix or variance–covariance matrix) is a matrix whose element in the i, j position is the covariance between the i-th and j-th elements of a random vector.

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Data compression

In signal processing, data compression, source coding, or bit-rate reduction involves encoding information using fewer bits than the original representation.

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Data mining

Data mining is the process of discovering patterns in large data sets involving methods at the intersection of machine learning, statistics, and database systems.

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Data set

A data set (or dataset) is a collection of data.

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David Hilbert

David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician.

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Deeplearning4j

Eclipse Deeplearning4j is a deep learning programming library written for Java and the Java virtual machine (JVM) and a computing framework with wide support for deep learning algorithms.

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Defective matrix

In linear algebra, a defective matrix is a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable.

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Degree of a polynomial

The degree of a polynomial is the highest degree of its monomials (individual terms) with non-zero coefficients.

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Degrees of freedom (mechanics)

In physics, the degree of freedom (DOF) of a mechanical system is the number of independent parameters that define its configuration.

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Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

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Determinant

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

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Diagonal

In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge.

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Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.

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Diagonalizable matrix

In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P−1AP is a diagonal matrix.

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Differential equation

A differential equation is a mathematical equation that relates some function with its derivatives.

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Differential operator

In mathematics, a differential operator is an operator defined as a function of the differentiation operator.

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Digital image processing

In computer science, Digital image processing is the use of computer algorithms to perform image processing on digital images.

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Direct sum

The direct sum is an operation from abstract algebra, a branch of mathematics.

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Discrete Laplace operator

In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete grid.

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Discriminant

In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them.

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Distributive property

In abstract algebra and formal logic, the distributive property of binary operations generalizes the distributive law from boolean algebra and elementary algebra.

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Dot product

In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.

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Earth Surface Processes and Landforms

Earth Surface Processes and Landforms is the journal of the British Society for Geomorphology (BSG), formerly the British Geomorphological Research Group (BGRG) and is an international journal of geomorphology, publishing on all aspects of Earth Surface Science.

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Eigendecomposition of a matrix

In linear algebra, eigendecomposition or sometimes spectral decomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors.

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Eigenface

Eigenfaces is the name given to a set of eigenvectors when they are used in the computer vision problem of human face recognition.

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Eigenfunction

In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue.

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Eigenplane

In mathematics, an eigenplane is a two-dimensional invariant subspace in a given vector space.

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Eigenvalue algorithm

In numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix.

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Eigenvector centrality

In graph theory, eigenvector centrality (also called eigencentrality) is a measure of the influence of a node in a network.

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Energy

In physics, energy is the quantitative property that must be transferred to an object in order to perform work on, or to heat, the object.

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Explained variation

In statistics, explained variation measures the proportion to which a mathematical model accounts for the variation (dispersion) of a given data set.

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Exponential function

In mathematics, an exponential function is a function of the form in which the argument occurs as an exponent.

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Facial recognition system

A facial recognition system is a technology capable of identifying or verifying a person from a digital image or a video frame from a video source.

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Factor analysis

Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors.

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Factorization

In mathematics, factorization (also factorisation in some forms of British English) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

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Finite element method

The finite element method (FEM), is a numerical method for solving problems of engineering and mathematical physics.

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Floating-point arithmetic

In computing, floating-point arithmetic is arithmetic using formulaic representation of real numbers as an approximation so as to support a trade-off between range and precision.

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Fock matrix

In the Hartree–Fock method of quantum mechanics, the Fock matrix is a matrix approximating the single-electron energy operator of a given quantum system in a given set of basis vectors.

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Formal power series

In mathematics, a formal power series is a generalization of a polynomial, where the number of terms is allowed to be infinite; this implies giving up the possibility of replacing the variable in the polynomial with an arbitrary number.

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Francesco Brioschi

Francesco Brioschi (22 December 1824 – 13 December 1897) was an Italian mathematician.

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Function space

In mathematics, a function space is a set of functions between two fixed sets.

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Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

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Fundamental theorem of algebra

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.

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Funk & Wagnalls

Funk & Wagnalls was an American publisher known for its reference works, including A Standard Dictionary of the English Language (1st ed. 1893-5), and the Funk & Wagnalls Standard Encyclopedia (25 volumes, 1st ed. 1912).

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Gaussian elimination

In linear algebra, Gaussian elimination (also known as row reduction) is an algorithm for solving systems of linear equations.

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Generalized eigenvector

In linear algebra, a generalized eigenvector of an n × n matrix A is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector.

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Geology

Geology (from the Ancient Greek γῆ, gē, i.e. "earth" and -λoγία, -logia, i.e. "study of, discourse") is an earth science concerned with the solid Earth, the rocks of which it is composed, and the processes by which they change over time.

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German language

German (Deutsch) is a West Germanic language that is mainly spoken in Central Europe.

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Google

Google LLC is an American multinational technology company that specializes in Internet-related services and products, which include online advertising technologies, search engine, cloud computing, software, and hardware.

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Graph theory

In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.

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Hamiltonian (quantum mechanics)

In quantum mechanics, a Hamiltonian is an operator corresponding to the total energy of the system in most of the cases.

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Hartree–Fock method

In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state.

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Heat equation

The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time.

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Henk van der Vorst

Hendrik "Henk" Albertus van der Vorst (born 5 May 1944, Venlo) is a Dutch mathematician and Emeritus Professor of Numerical Analysis at Utrecht University.

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Henri Poincaré

Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science.

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Hermann Schwarz

Karl Hermann Amandus Schwarz (25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis.

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Hermann von Helmholtz

Hermann Ludwig Ferdinand von Helmholtz (August 31, 1821 – September 8, 1894) was a German physician and physicist who made significant contributions in several scientific fields.

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Hermitian matrix

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and: Hermitian matrices can be understood as the complex extension of real symmetric matrices.

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Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

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Houghton Mifflin Harcourt

Houghton Mifflin Harcourt (HMH) is an educational and trade publisher in the United States.

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Householder transformation

In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin.

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Identity matrix

In linear algebra, the identity matrix, or sometimes ambiguously called a unit matrix, of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere.

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Inertia

Inertia is the resistance of any physical object to any change in its position and state of motion.

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Integral transform

In mathematics, an integral transform maps an equation from its original domain into another domain where it might be manipulated and solved much more easily than in the original domain.

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Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that if a continuous function, f, with an interval,, as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

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Introduction to eigenstates

Because of the uncertainty principle, statements about both the position and momentum of particles can only assign a probability that the position or momentum will have some numerical value.

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Invariant subspace

In mathematics, an invariant subspace of a linear mapping T: V → V from some vector space V to itself is a subspace W of V that is preserved by T; that is, T(W) ⊆ W.

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Inverse iteration

In numerical analysis, inverse iteration (also known as the inverse power method) is an iterative eigenvalue algorithm.

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Invertible matrix

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate) if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.

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Ionization energy

The ionization energy (Ei) is qualitatively defined as the amount of energy required to remove the most loosely bound electron, the valence electron, of an isolated gaseous atom to form a cation.

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Irrational number

In mathematics, the irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios (or fractions) of integers.

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Iteration

Iteration is the act of repeating a process, to generate a (possibly unbounded) sequence of outcomes, with the aim of approaching a desired goal, target or result.

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Iterative method

In computational mathematics, an iterative method is a mathematical procedure that uses an initial guess to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones.

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Jacques Charles François Sturm

Jacques Charles François Sturm ForMemRS (29 September 1803 – 15 December 1855) was a French mathematician.

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James Demmel

James Weldon Demmel is an American mathematician and computer scientist, the Dr.

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Johann Andreas Segner

Johann Segner (János András Segner, Johann Andreas von Segner, Ján Andrej Segner, Iohannes Andreas de Segner; October 9, 1704 – October 5, 1777) was a Hungarian scientist.

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John G. F. Francis

John G.F. Francis (born 1934) is an English computer scientist, who in 1961 published the QR algorithm for computing the eigenvalues and eigenvectors of matrices, which has been named as one of the ten most important algorithms of the twentieth century.

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John Wiley & Sons

John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.

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Jordan normal form

In linear algebra, a Jordan normal form (often called Jordan canonical form) of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis.

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Joseph Fourier

Jean-Baptiste Joseph Fourier (21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations.

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Joseph Liouville

Joseph Liouville FRS FRSE FAS (24 March 1809 – 8 September 1882) was a French mathematician.

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Joseph-Louis Lagrange

Joseph-Louis Lagrange (or;; born Giuseppe Lodovico Lagrangia, Encyclopædia Britannica or Giuseppe Ludovico De la Grange Tournier, Turin, 25 January 1736 – Paris, 10 April 1813; also reported as Giuseppe Luigi Lagrange or Lagrangia) was an Italian Enlightenment Era mathematician and astronomer.

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Karl Weierstrass

Karl Theodor Wilhelm Weierstrass (Weierstraß; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis".

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Kernel (linear algebra)

In mathematics, and more specifically in linear algebra and functional analysis, the kernel (also known as null space or nullspace) of a linear map between two vector spaces V and W, is the set of all elements v of V for which, where 0 denotes the zero vector in W. That is, in set-builder notation,.

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Koopmans' theorem

Koopmans' theorem states that in closed-shell Hartree–Fock theory (HF), the first ionization energy of a molecular system is equal to the negative of the orbital energy of the highest occupied molecular orbital (HOMO).

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Lanczos algorithm

The Lanczos algorithm is a direct algorithm devised by Cornelius Lanczos that is an adaptation of power methods to find the m most useful eigenvalues and eigenvectors of an n \times n Hermitian matrix, where m is often but not necessarily much smaller than n. Although computationally efficient in principle, the method as initially formulated was not useful, due to its numerical instability.

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Laplace's equation

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties.

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Laplacian matrix

In the mathematical field of graph theory, the Laplacian matrix, sometimes called admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph.

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Leibniz formula for determinants

In algebra, the Leibniz formula, named in honor of Gottfried Leibniz, expresses the determinant of a square matrix in terms of permutations of the matrix elements.

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Leonhard Euler

Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.

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Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.

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Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

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Linear equation

In mathematics, a linear equation is an equation that may be put in the form where x_1, \ldots, x_n are the variables or unknowns, and c, a_1, \ldots, a_n are coefficients, which are often real numbers, but may be parameters, or even any expression that does not contain the unknowns.

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Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

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Linear subspace

In linear algebra and related fields of mathematics, a linear subspace, also known as a vector subspace, or, in the older literature, a linear manifold, is a vector space that is a subset of some other (higher-dimension) vector space.

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Linear system

A linear system is a mathematical model of a system based on the use of a linear operator.

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List of numerical analysis software

Listed here are end-user computer applications intended for use with numerical or data analysis.

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Marketing

Marketing is the study and management of exchange relationships.

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Markov chain

A Markov chain is "a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event".

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Mass matrix

In analytical mechanics, the mass matrix is a symmetric matrix M that expresses the connection between the time derivative \dot q of the generalized coordinate vector q of a system and the kinetic energy T of that system, by the equation where \dot q^\mathrm denotes the transpose of the vector \dot q. This equation is analogous to the formula for the kinetic energy of a particle with mass m and velocity v, namely and can be derived from it, by expressing the position of each particle of the system in terms of q. In general, the mass matrix M depends on the state q, and therefore varies with time.

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MathWorld

MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein.

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Matrix (mathematics)

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

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Matrix decomposition

In the mathematical discipline of linear algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices.

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Matrix multiplication

In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.

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Matrix similarity

In linear algebra, two n-by-n matrices and are called similar if for some invertible n-by-n matrix.

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Mechanics

Mechanics (Greek μηχανική) is that area of science concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment.

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Module (mathematics)

In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.

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Molecular orbital

In chemistry, a molecular orbital (MO) is a mathematical function describing the wave-like behavior of an electron in a molecule.

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Molecular physics

Molecular physics is the study of the physical properties of molecules, the chemical bonds between atoms as well as the molecular dynamics.

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Moment of inertia

The moment of inertia, otherwise known as the angular mass or rotational inertia, of a rigid body is a tensor that determines the torque needed for a desired angular acceleration about a rotational axis; similar to how mass determines the force needed for a desired acceleration.

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Mona Lisa

The Mona Lisa (Monna Lisa or La Gioconda, La Joconde) is a half-length portrait painting by the Italian Renaissance artist Leonardo da Vinci that has been described as "the best known, the most visited, the most written about, the most sung about, the most parodied work of art in the world".

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Multiplicity (mathematics)

In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset.

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Multivariate statistics

Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable.

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Nonlinear eigenproblem

A nonlinear eigenproblem is a generalization of an ordinary eigenproblem to equations that depend nonlinearly on the eigenvalue.

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Numerical method

In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems.

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Observable

In physics, an observable is a dynamic variable that can be measured.

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Orthogonal basis

In mathematics, particularly linear algebra, an orthogonal basis for an inner product space is a basis for whose vectors are mutually orthogonal.

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Orthogonal matrix

In linear algebra, an orthogonal matrix is a square matrix whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e. where I is the identity matrix.

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Orthogonality

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.

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PageRank

PageRank (PR) is an algorithm used by Google Search to rank websites in their search engine results.

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Parallel (geometry)

In geometry, parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel.

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Permutation matrix

\pi.

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Pixel

In digital imaging, a pixel, pel, dots, or picture element is a physical point in a raster image, or the smallest addressable element in an all points addressable display device; so it is the smallest controllable element of a picture represented on the screen.

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Poisson's equation

In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics.

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Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

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Polynomial long division

In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalised version of the familiar arithmetic technique called long division.

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Positive-definite matrix

In linear algebra, a symmetric real matrix M is said to be positive definite if the scalar z^Mz is strictly positive for every non-zero column vector z of n real numbers.

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Power iteration

In mathematics, power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix A, the algorithm will produce a number \lambda, which is the greatest (in absolute value) eigenvalue of A, and a nonzero vector v, the corresponding eigenvector of \lambda, such that Av.

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Principal component analysis

Principal component analysis (PCA) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components.

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Psychometrics

Psychometrics is a field of study concerned with the theory and technique of psychological measurement.

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Q methodology

Q Methodology is a research method used in psychology and in social sciences to study people's "subjectivity"—that is, their viewpoint.

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QR algorithm

In numerical linear algebra, the QR algorithm is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix.

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Quadratic eigenvalue problem

In mathematics, the quadratic eigenvalue problem (QEP), is to find scalar eigenvalues \lambda, left eigenvectors y and right eigenvectors x such that where Q(\lambda).

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Quadratic equation

In algebra, a quadratic equation (from the Latin quadratus for "square") is any equation having the form where represents an unknown, and,, and represent known numbers such that is not equal to.

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Quadratic form

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables.

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Quadric

In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas).

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Quantum chemistry

Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems.

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Quantum mechanics

Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.

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Quantum state

In quantum physics, quantum state refers to the state of an isolated quantum system.

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Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

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Reciprocal polynomial

In algebra, the reciprocal polynomial, or reflected polynomial* or, of a polynomial of degree with coefficients from an arbitrary field, such as is the polynomial Essentially, the coefficients are written in reverse order.

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Recurrence relation

In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequence or array is defined as a function of the preceding terms.

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Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.

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Richard von Mises

Richard Edler von Mises (19 April 1883 – 14 July 1953) was a scientist and mathematician who worked on solid mechanics, fluid mechanics, aerodynamics, aeronautics, statistics and probability theory.

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Rigid body

In physics, a rigid body is a solid body in which deformation is zero or so small it can be neglected.

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Roothaan equations

The Roothaan equations are a representation of the Hartree–Fock equation in a non orthonormal basis set which can be of Gaussian-type or Slater-type.

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Rotation (mathematics)

Rotation in mathematics is a concept originating in geometry.

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Round-off error

A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value due to rounding.

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Row and column vectors

In linear algebra, a column vector or column matrix is an m × 1 matrix, that is, a matrix consisting of a single column of m elements, Similarly, a row vector or row matrix is a 1 × m matrix, that is, a matrix consisting of a single row of m elements Throughout, boldface is used for the row and column vectors.

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Scalar (mathematics)

A scalar is an element of a field which is used to define a vector space.

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Scalar multiplication

In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra).

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Scaling (geometry)

In Euclidean geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions.

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Schrödinger equation

In quantum mechanics, the Schrödinger equation is a mathematical equation that describes the changes over time of a physical system in which quantum effects, such as wave–particle duality, are significant.

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Self-adjoint operator

In mathematics, a self-adjoint operator on a finite-dimensional complex vector space V with inner product \langle\cdot,\cdot\rangle is a linear map A (from V to itself) that is its own adjoint: \langle Av,w\rangle.

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Separation of variables

In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.

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Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

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Shear mapping

In plane geometry, a shear mapping is a linear map that displaces each point in fixed direction, by an amount proportional to its signed distance from a line that is parallel to that direction.

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Singular value

In mathematics, in particular functional analysis, the singular values, or s-numbers of a compact operator acting between Hilbert spaces X and Y, are the square roots of the eigenvalues of the non-negative self-adjoint operator (where T* denotes the adjoint of T).

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Skew-symmetric matrix

In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative; that is, it satisfies the condition In terms of the entries of the matrix, if aij denotes the entry in the and; i.e.,, then the skew-symmetric condition is For example, the following matrix is skew-symmetric: 0 & 2 & -1 \\ -2 & 0 & -4 \\ 1 & 4 & 0\end.

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Solid mechanics

Solid mechanics is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and other external or internal agents.

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Sparse matrix

In numerical analysis and computer science, a sparse matrix or sparse array is a matrix in which most of the elements are zero.

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Spectral clustering

In multivariate statistics and the clustering of data, spectral clustering techniques make use of the spectrum (eigenvalues) of the similarity matrix of the data to perform dimensionality reduction before clustering in fewer dimensions.

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Spectral graph theory

In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix.

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Spectrum (functional analysis)

In mathematics, particularly in functional analysis, the spectrum of a bounded operator is a generalisation of the set of eigenvalues of a matrix.

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Square matrix

In mathematics, a square matrix is a matrix with the same number of rows and columns.

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Square-integrable function

In mathematics, a square-integrable function, also called a quadratically integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite.

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Squeeze mapping

In linear algebra, a squeeze mapping is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapping.

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Stability theory

In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions.

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Stationary distribution

Stationary distribution may refer to.

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Statistical hypothesis testing

A statistical hypothesis, sometimes called confirmatory data analysis, is a hypothesis that is testable on the basis of observing a process that is modeled via a set of random variables.

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Statistical significance

In statistical hypothesis testing, a result has statistical significance when it is very unlikely to have occurred given the null hypothesis.

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Stiffness matrix

In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix represents the system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation.

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Stress (mechanics)

In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighboring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material.

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Structural equation modeling

Structural equation modeling (SEM) includes a diverse set of mathematical models, computer algorithms, and statistical methods that fit networks of constructs to data.

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Sturm–Liouville theory

In mathematics and its applications, a classical Sturm–Liouville theory, named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), is the theory of a real second-order linear differential equation of the form where y is a function of the free variable x. Here the functions p(x), q(x), and w(x) > 0 are specified at the outset.

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Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.

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Tensor

In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors.

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Till

Closeup of glacial till. Note that the larger grains (pebbles and gravel) in the till are completely surrounded by the matrix of finer material (silt and sand), and this characteristic, known as ''matrix support'', is diagnostic of till. Glacial till with tufts of grass Till or glacial till is unsorted glacial sediment.

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Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e., where aii denotes the entry on the ith row and ith column of A. The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis.

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Triangular matrix

In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix.

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Turn (geometry)

A turn is a unit of plane angle measurement equal to 2pi radians, 360 degrees or 400 gradians.

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Union (set theory)

In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.

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Unit circle

In mathematics, a unit circle is a circle with a radius of one.

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Unitary matrix

In mathematics, a complex square matrix is unitary if its conjugate transpose is also its inverse—that is, if where is the identity matrix.

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University of Nottingham

The University of Nottingham is a public research university in Nottingham, United Kingdom.

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Variance

In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean.

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Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

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Vera Kublanovskaya

Vera Nikolaevna Kublanovskaya (nee Totubalina; November 21, 1920 – February 21, 2012) was a Russian mathematician noted for her work on developing computational methods for solving spectral problems of algebra.

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Vibration

Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point.

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Wave function

A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system.

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Weight (representation theory)

In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one-dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group.

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Wilkinson's polynomial

In numerical analysis, Wilkinson's polynomial is a specific polynomial which was used by James H. Wilkinson in 1963 to illustrate a difficulty when finding the root of a polynomial: the location of the roots can be very sensitive to perturbations in the coefficients of the polynomial.

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Zero element

In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures.

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References

[1] https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

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