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Method of analysis in probability theory
the matrix geometric method is a method for the analysis of quasi-birth–death processes, continuous-time Markov chain whose transition rate matrix has
Matrix_geometric_method
Topics referred to by the same term
analysis in geometric optics, a mathematical method for performing ray tracing calculations. Transfer-matrix method (combinatorics), a method for computing
Transfer-matrix_method
Computing technique in probability theory
in an M/G/1 queue. The method is a more complicated version of the matrix geometric method and is the classical solution method for M/G/1 chains. An M/G/1-type
Matrix_analytic_method
Math operation methods
A {\displaystyle A^{-1}A} is the identity matrix. There are many methods for calculating an inverse matrix, if it exists. Gaussian elimination is a useful
Methods_of_matrix_inversion
Matrix decomposition
(also known as eigenvalue decomposition or EVD) is a factorization of a matrix A {\displaystyle A} into a canonical form given by A = Q D Q − 1 {\displaystyle
Eigendecomposition of a matrix
Eigendecomposition_of_a_matrix
Concepts from linear algebra
could be for a matrix with two distinct eigenvalues. Geometric multiplicities are defined in a later section. For a Hermitian matrix A, the norm squared
Eigenvalues_and_eigenvectors
Mathematical function of two positive real arguments
arithmetic–geometric mean (AGM or agM) of two positive real numbers x and y is the mutual limit of a sequence of arithmetic means and a sequence of geometric means
Arithmetic–geometric_mean
Belgian-American mathematician (1935–2014)
York, NY: Dekker. ISBN 978-0-8247-8283-2. Neuts, Marcel F. (1994). Matrix-geometric solutions in stochastic models: an algorithmic approach. Dover books
Marcel_F._Neuts
Square matrix with ones on the main diagonal and zeros elsewhere
elsewhere. It has unique properties; for example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation
Identity_matrix
Square matrix containing the distances between elements in a set
character is encoded in the geometric-distance matrix. The geometric-distance matrix is a different type of distance matrix that is based on the graph-theoretical
Distance_matrix
Matrix of second derivatives
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function
Hessian_matrix
Mathematical study of waiting lines, or queues
in the ARPANET, a forerunner to the Internet. The matrix geometric method and matrix analytic methods have allowed queues with phase-type distributed inter-arrival
Queueing_theory
Array of numbers
attached to matrices (see above). Another matrix frequently used in geometrical situations is the Jacobi matrix of a differentiable map f : R n → R m
Matrix_(mathematics)
Arithmetic mean is greater than or equal to geometric mean
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative
AM–GM_inequality
Phase of a cycle
In classical and quantum mechanics, the geometric phase (also known as the Pancharatnam–Berry phase, Pancharatnam phase, or Berry phase) is a phase difference
Geometric_phase
Matrix of partial derivatives of a vector-valued function
vector calculus, the Jacobian matrix (/dʒəˈkoʊbiən/, /dʒɪ-, jɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial
Jacobian matrix and determinant
Jacobian_matrix_and_determinant
Central object in linear algebra; mapping vectors to vectors
there exists an m × n {\displaystyle m\times n} matrix A {\displaystyle A} , called the transformation matrix of T {\displaystyle T} , such that: T ( x )
Transformation_matrix
Algorithm for solving systems of linear equations
corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse
Gaussian_elimination
N-th root of the product of n numbers
In mathematics, the geometric mean (also known as the mean proportional) is a mean or average which indicates a central tendency of a finite collection
Geometric_mean
Algorithm for linear programming
question are the corners (i.e., the neighborhoods of the vertices) of a geometric object called a polytope. The shape of this polytope is defined by the
Simplex_algorithm
Matrices similar to diagonal matrices
P D P − 1 {\displaystyle A=PDP^{-1}} . The geometric transformation represented by a diagonalizable matrix is an inhomogeneous dilation (or anisotropic
Diagonalizable_matrix
Mathematical operation in linear algebra
columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix product, has the number
Matrix_multiplication
Mathematics concept
whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows in a Hadamard matrix represents two perpendicular vectors, while
Hadamard_matrix
Stochastic matrix representing links between entities
iteratively from the Google matrix using the power method. However, in order for the power method to converge, the matrix must be stochastic, irreducible
Google_matrix
Matrix of geometric progressions
linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an
Vandermonde_matrix
Method for finding stationary points of a function
x_{k+1}=x_{k}+t=x_{k}-{\frac {f'(x_{k})}{f''(x_{k})}}.} The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting
Newton's method in optimization
Newton's_method_in_optimization
geometric interpretation from the use of screw axes for each joint. The POE method was introduced by Roger W. Brockett in 1984. The following method is
Product of exponentials formula
Product_of_exponentials_formula
Sum of an (infinite) geometric progression
for matrix-valued geometric series, function-valued geometric series, p {\displaystyle p} -adic number geometric series, and most generally geometric series
Geometric_series
Matrix representing a Euclidean rotation
rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix R = [
Rotation_matrix
Probability distribution
section. § Geometric variance and covariance contains plots and further discussion of the Fisher information matrix components: the log geometric variances
Beta_distribution
Algebraic structure designed for geometry
geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. Geometric algebra
Geometric_algebra
Geometric modeling kernel
Russian Geometric Kernel (also known as RGK) is a proprietary geometric modeling kernel developed by several Russian software companies, most notably
Russian_Geometric_Kernel
Orthonormalization of a set of vectors
{\displaystyle k} the proof is accomplished by mathematical induction. Geometrically, this method proceeds as follows: to compute u i {\displaystyle \mathbf {u}
Gram–Schmidt_process
Measure of covariance of components of a random vector
covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the
Covariance_matrix
distribution of a quasi-birth-death process can be computed using the matrix geometric method. Latouche, G. (2011). "Level-Independent Quasi-Birth-and-Death
Quasi-birth–death_process
Matrix with a multiplicative inverse
n} matrix is invertible if its rank is n {\displaystyle n} , by the rank-nullity theorem. Such a matrix is said to be of full rank. Geometrically, this
Invertible_matrix
Property of a mathematical matrix
In mathematics, a symmetric matrix M {\displaystyle M} with real entries is positive-definite if the real number x T M x {\displaystyle \mathbf {x} ^{\mathsf
Definite_matrix
Method of digital signal processing
Geometric phase analysis is a method of digital signal processing used to determine crystallographic quantities such as d-spacing or strain from high-resolution
Geometric_phase_analysis
Multi-criteria decision analysis method
is a method of compensatory aggregation that compares a set of alternatives, normalising scores for each criterion, and calculating the geometric distance
TOPSIS
Method of data analysis
explicitly calculating and storing the covariance matrix XTX, instead utilizing one of matrix-free methods, for example, based on the function evaluating
Principal_component_analysis
Concept in linear algebra
{\vec {v}}} is normal to. In geometric optics, specular reflection can be expressed in terms of the Householder matrix (see Specular reflection § Vector
Householder_transformation
Mathematical element in composite engineering
the volume fraction of both the fiber and matrix in the original laminate may be determined. This method is generally used for composites composed of
Fiber_volume_ratio
Equivalence under a change of basis (linear algebra)
multiplicities Geometric multiplicities of eigenvalues (but not the eigenspaces, which are transformed according to the base change matrix P used). Minimal
Matrix_similarity
Frobenius inner product Frobenius norm Frobenius manifold Frobenius matrix Frobenius method Frobenius normal form Frobenius polynomial Frobenius pseudoprime
List of things named after Ferdinand Georg Frobenius
List_of_things_named_after_Ferdinand_Georg_Frobenius
Matrix decomposition
complex matrix into a rotation, followed by a scaling, followed by another rotation. It generalizes the eigendecomposition of a square normal matrix with
Singular_value_decomposition
Mathematical sequence satisfying a specific pattern
mathematics, an arithmetico-geometric sequence is the result of element-by-element multiplication of the elements of a geometric progression with the corresponding
Arithmetico-geometric sequence
Arithmetico-geometric_sequence
Correspondence between quaternions and 3D rotations
second rotation. This is a geometric proof that conjugation by q and by −q must produce the same rotational transformation matrix. That fact is confirmed
Quaternions and spatial rotation
Quaternions_and_spatial_rotation
Formula for systems of linear equations
expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector
Cramer's_rule
Algorithm for finding zeros of functions
{\displaystyle D^{2}f} is the 2nd derivative Hessian matrix). Newton's method is one of many known methods of computing square roots. Given a positive number
Newton's_method
Means of solving M/M/c queue models in queueing theory
a class of Markov models: Application and comparison with the matrix-geometric method". Performance Evaluation. 23 (3): 241. doi:10.1016/0166-5316(94)00025-F
Spectral_expansion_solution
Method of solving differential equations
extremely diverse as they include Krylov subspace methods and can be preconditioned. Any geometric multigrid cycle iteration is performed on a hierarchy
Multigrid_method
Process by which dust, particulates, etc. scatter light
evolved. The technique is also known as null field method and extended boundary technique method (EBCM). Matrix elements are obtained by matching boundary conditions
Light_scattering_by_particles
Ways to represent 3D rotations
devices. Using complimentary filter (popular alternative to Kalman filter) with DCM matrix. The Turn Sphere – A geometric method of composing two rotations.
Rotation formulations in three dimensions
Rotation_formulations_in_three_dimensions
Branch of mathematics
related methods. Fundamental matrix (computer vision) Geometric algebra Linear programming Linear regression, a statistical estimation method Numerical
Linear_algebra
Algorithm used for frequency estimation and radio direction finding
signal or autocorrelation matrix using an eigenspace method. Since R x {\displaystyle \mathbf {R} _{x}} is a Hermitian matrix, all of its M {\displaystyle
MUSIC_(algorithm)
Infinite sum
and squaring method for the matrix exponential revisited. SIAM review, 51(4), 747-764. How and How Not to Compute the Exponential of a Matrix Nicolas Bourbaki
Series_(mathematics)
Data analysis technique
the indicator matrix allows the direct representation of individuals as points in geometric space. The Burt table is the symmetric matrix of all two-way
Multiple correspondence analysis
Multiple_correspondence_analysis
System for describing optical polarization
Mueller calculus is a matrix method for manipulating Stokes vectors, which represent the polarization of light. It was developed in 1943 by Hans Mueller
Mueller_calculus
Statistical technique
in the Frobenius norm, low-rank approximation of the data matrix. In the least squares method of data modeling, the objective function to be minimized
Total_least_squares
Several equations of degree 1 to be solved simultaneously
definite matrix can be solved twice as fast with the Cholesky decomposition. Levinson recursion is a fast method for Toeplitz matrices. Special methods exist
System_of_linear_equations
Representation of a matrix as a product
algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions;
Matrix_decomposition
Semiempirical quantum chemistry method
closely similar method was used earlier by Hoffmann and William Lipscomb for studies of boron hydrides. The off-diagonal Hamiltonian matrix elements were
Extended_Hückel_method
Branch of physics
finite element methods); matrix products (when using transfer matrix methods); calculating numerical integrals (when using the method of moments); using
Computational electromagnetics
Computational_electromagnetics
Matrix decomposition
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
QR_decomposition
Specialized notation for multivariable calculus
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various
Matrix_calculus
Matrix decomposition method
decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for
Cholesky_decomposition
computational geometry. Geometric function theory the study of geometric properties of analytic functions. Geometric invariant theory a method for constructing
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Quantitative study of size and shape
JSTOR 2992207. Walker, J. (2000). "The ability of geometric morphometric methods to estimate a known covariance matrix". Systematic Biology. 49 (4): 686–696. doi:10
Morphometrics
Most widely known generalized inverse of a matrix
A^{+}} of a matrix A {\displaystyle A} , often called the pseudoinverse, is the most widely known generalization of the inverse matrix. It was independently
Moore–Penrose_inverse
analysis: Sparse matrix Band matrix Bidiagonal matrix Tridiagonal matrix Pentadiagonal matrix Skyline matrix Circulant matrix Triangular matrix Diagonally dominant
List of numerical analysis topics
List_of_numerical_analysis_topics
Ray tracing technique
case for all rays must still focus the paraxial rays correctly, this matrix method will properly describe the positions of focal planes and magnifications
Ray_transfer_matrix_analysis
Creation of a 3D model from a set of images
equations due to Kruppa, which are derived from a geometric interpretation of the rigidity constraint. The matrix K = A A ⊤ {\displaystyle K=AA^{\top }} is unknown
3D reconstruction from multiple images
3D_reconstruction_from_multiple_images
Algorithm
problem for which the selection method in will perform in an inferior manner. The Kaczmarz iteration (1) has a purely geometric interpretation: the algorithm
Kaczmarz_method
Mathematical inequality
Franklin Muirhead, also known as the "bunching" method, generalizes the inequality of arithmetic and geometric means. For any real vector a = ( a 1 , … , a
Muirhead's_inequality
Mathematical operation on vectors in 3D space
vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional
Cross_product
Process of estimating the parameters of a pinhole camera model
the camera parameters are represented in a 3 × 4 projection matrix called the camera matrix. The extrinsic parameters define the camera pose (position
Camera_resectioning
Notions of sums for matrices in linear algebra
{\displaystyle {\vec {v}}\!} , adding two matrices would have the geometric effect of applying each matrix transformation separately onto v → {\displaystyle {\vec
Matrix_addition
In mathematics, invariant of square matrices
the determinant is a scalar-valued function of the entries of a square matrix that has many properties which make it fundamental for the study of square
Determinant
Numerical methods for matrix eigenvalue calculation
finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors. Given an n × n square matrix A of real or complex numbers
Eigenvalue_algorithm
Field of mathematics
and computational statistics. Matrix methods are particularly used in finite difference methods, finite element methods, and the modeling of differential
Numerical_linear_algebra
Evolutionary algorithm
are represented by a covariance matrix. The covariance matrix adaptation (CMA) is a method to update the covariance matrix of this distribution. This is
CMA-ES
Many of these approximation methods can be expressed in purely linear algebraic or functional analytic terms as matrix or function approximations. Others
Gaussian process approximations
Gaussian_process_approximations
Vectors mapped to 0 by a linear map
the n × n identity matrix. Computing its column echelon form by Gaussian elimination (or any other suitable method), we get a matrix [ B C ] . {\displaystyle
Kernel_(linear_algebra)
Area of discrete mathematics
the degree of a vertex) and the adjacency matrix. Group theory, particularly automorphism groups and geometric group theory, focuses on various families
Graph_theory
Notion in statistics
1111/j.2517-6161.1987.tb01422.x. Watanabe, S. (2008), "Algebraic geometrical method in singular statistical estimation", in Accardi, L.; Freudenberg,
Fisher_information
Numerical method for solving physical or engineering problems
developed by combining mesh-free methods with the finite element method. Spectral element methods combine the geometric flexibility of finite elements and
Finite_element_method
On converting relations to functions of several real variables
{\displaystyle (Df)(a,b)=\left[{\begin{matrix}-1&\cdots &0\\\vdots &\ddots &\vdots \\0&\cdots &-1\end{matrix}}\left|{\begin{matrix}{\frac {\partial h_{1}}{\partial
Implicit_function_theorem
Nonlinear dimensionality reduction method
algorithm, for example). The top n eigenvectors of the geodesic distance matrix, represent the coordinates in the new n-dimensional Euclidean space. A very
Isomap
Method of differentiating single-term polynomials
{p}{q}}-a^{\frac {p}{q}}}{b-a}}\\[4pt]\end{aligned}}} Now, consider the geometric sum formula, b n − a n b − a = ∑ i = 0 n − 1 b ( n − 1 ) − i a i {\displaystyle
Power_rule
Algebraic object with geometric applications
matrix and its inverse cancel, so that expressions like v i e i {\displaystyle {v}^{i}\,\mathbf {e} _{i}} can immediately be seen to be geometrically
Tensor
Technology, 2012 Anil Damle, Yuekai Sun: A geometric approach to archetypal analysis and non-negative matrix factorization. arXiv preprint: arXiv : 1405
Archetypal_analysis
Non-tensorial representation of the spin group
associated with Euclidean space. Spinors can be thought of as companion geometric objects to Euclidean space that, like Euclidean vectors, respond when
Spinor
Eigenvalue algorithm
power iteration (also known as the power method) is an eigenvalue algorithm: given a diagonalizable matrix A {\displaystyle A} , the algorithm will produce
Power_iteration
Robotics problem on coordinating two parts of a robot
method for hand–eye calibration." 19 July 2017. Irene Fassi, Giovanni Legnani "Hand to sensor calibration: A geometrical interpretation of the matrix
Hand–eye_calibration_problem
Successive over-relaxation (SOR): method used to speed up convergence of the Gauss–Seidel method Tridiagonal matrix algorithm (Thomas algorithm): solves
List_of_algorithms
Computer-aided geometric design
approximation method is an iterative method of data fitting with geometric meanings. Given a set of data points to be fitted, the method obtains a series
Progressive-iterative approximation method
Progressive-iterative_approximation_method
Algebraic operation on coordinate vectors
products of the corresponding entries of the two sequences of numbers. Geometrically, the scalar product of two vectors is the product of their lengths and
Dot_product
Description of the orientation of a rigid body
rotation matrix E.g. Appendix I (p. 483) of: Roithmayr, Carlos M.; Hodges, Dewey H. (2016). Dynamics: Theory and Application of Kane's Method (1st ed.)
Euler_angles
Method used in early text mode video hardware to emulate raster graphics
all binary combinations of a certain subdivision matrix of the text mode character size; this method is referred to as block graphics, or sometimes mosaic
Semigraphics
Every square matrix with positive entries can be written in a certain standard form
first matrix by a positive number and dividing the second one by the same number. A simple iterative method to approach the double stochastic matrix is to
Sinkhorn's_theorem
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