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Relation of a matrix of variables between two points in time
In mathematics, a matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point
Matrix_difference_equation
Pattern defining an infinite sequence of numbers
and "difference equation" can be used interchangeably. See Rational difference equation, Linear constant-coefficient difference equation and Matrix difference
Recurrence_relation
Matrix whose entries are the coefficients of a linear equation
algebra, a coefficient matrix is a matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems
Coefficient_matrix
Type of mathematical equation
derivatives of various orders. A matrix differential equation contains more than one function stacked into vector form with a matrix relating the functions to
Matrix_differential_equation
Cauchy–Riemann equations Cesàro equation Chapman–Kolmogorov equation Chebyshev equation Chrystal's equation Chudnovsky formula Clairaut's equation Clausen's
List_of_equations
Mathematical relation defining a sequence
coefficients (also known as a linear recurrence relation or linear difference equation) sets equal to 0 a polynomial that is linear in the various iterates
Linear recurrence with constant coefficients
Linear_recurrence_with_constant_coefficients
Class of numerical techniques
finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating derivatives with finite differences.
Finite_difference_method
Matrix used to describe the transitions of a Markov chain
𝝉(I−T)−nTn−1𝟏. Density matrix Markov kernel, the equivalent of a stochastic matrix over a continuous state space Matrix difference equation Models of DNA evolution
Stochastic_matrix
In mathematics, a rational difference equation is a nonlinear difference equation of the form xn+1=α+∑i=0kβixn−iA+∑i=0kBixn−i , where the initial conditions
Rational_difference_equation
Polynomial whose roots are the eigenvalues of a matrix
is represented by a square matrix A. Then an eigenvector 𝐯 and the corresponding eigenvalue λ must satisfy the equation A𝐯=λ𝐯, or, equivalently (since
Characteristic_polynomial
Array of numbers
matrix and the Fock matrix used in solving the Roothaan equations to obtain the molecular orbitals of the Hartree–Fock method. The adjacency matrix of
Matrix_(mathematics)
Several equations of degree 1 to be solved simultaneously
otherwise not guaranteed. The vector equation is equivalent to a matrix equation of the form A𝐱=𝐛 where A is an m×n matrix, x is a column vector with n entries
System_of_linear_equations
Representation of a matrix as a sum
differential equations) depend upon the direct solution of matrix equations involving matrices more general than tridiagonal matrices. These matrix equations can
Matrix_splitting
Type of differential equation
an equation of this form (1694) and Jacob Bernoulli obtained a solution for a special case (1703). The term Riccati equation is also used for matrix equations
Riccati_equation
Matrix formed by appending columns of two other matrices
row operations on the augmented matrix (A|B) as is done on the original one A when solving a system of linear equations by Gaussian elimination. For example
Augmented_matrix
Matrix operation generalizing exponentiation of scalar numbers
of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding
Matrix_exponential
Concepts from linear algebra
characteristic equation λk−a1λk−1−a2λk−2−⋯−ak−1λ−ak=0, which can be found by stacking into matrix form a set of equations consisting of the above difference equation
Eigenvalues_and_eigenvectors
Nonlinear equation which arises on linear optimal control problems
invariant solutions of the matrix valued Riccati difference equation (which is the analogue of the Riccati differential equation in the context of discrete
Algebraic_Riccati_equation
Mathematical tool in quantum physics
In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed
Density_matrix
Formulation of quantum mechanics
frequency. To determine the matrix elements, Heisenberg required that the classical equations of motion be obeyed as matrix equations, dXdt=P ,dPdt=−X−3εX2
Matrix_mechanics
For a square matrix, the transpose of the cofactor matrix
fundamental results in matrix algebra, that A is invertible if and only if det(A) is an invertible element of R. When this holds, the equation above yields
Adjugate_matrix
Discrete analog of a derivative
Δ[f](x)=f(x+1)−f(x). A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves
Finite_difference
Partial differential equation describing the evolution of temperature in a region
The heat equation is a parabolic partial differential equation that occurs in the theory of heat transfer and models the evolution of the temperature
Heat_equation
Relativistic quantum mechanical wave equation
significantly across dimensions. Other differences include the absence of a chirality matrix in odd dimensions. The equation can also be generalized from flat
Dirac_equation
Type of differential equation
differential equations List of dynamical systems and differential equations topics Matrix differential equation Numerical partial differential equations Partial
Partial_differential_equation
Set of quasilinear hyperbolic equations governing adiabatic and inviscid flow
In fluid dynamics, the Euler equations are a set of partial differential equations governing adiabatic and inviscid flow. They are named after Leonhard
Euler equations (fluid dynamics)
Euler_equations_(fluid_dynamics)
Coefficient used in numerical approximation
the number of stencil points, the finite difference coefficients can be obtained by solving the linear equations ( s 1 0 ⋯ s N 0 ⋮ ⋱ ⋮ s 1 N − 1 ⋯ s N N
Finite_difference_coefficient
Mandelbrot set Recurrence relation Matrix difference equation Rational difference equation Examples of differential equations Autonomous system (mathematics)
List of dynamical systems and differential equations topics
List_of_dynamical_systems_and_differential_equations_topics
Method for estimating the unknown parameters in a linear regression model
p columns of the matrix 𝐗 are linearly independent, given by solving the so-called normal equations: (𝐗T𝐗)𝜷̂=𝐗T𝐲 . The matrix 𝐗T𝐗 is known as
Ordinary_least_squares
Mathematical description of fluid movements
the endothelial cells. This fibre matrix endocapillary layer is called the endothelial glycocalyx. The Starling equation describes that relationship in mathematical
Starling_equation
Limiting set in dynamical systems
Likewise, a linear matrix difference equation in a dynamic vector X, of the homogeneous form Xt=AXt−1 in terms of square matrix A will have all elements
Attractor
Age-structured model of population growth
polynomial of the matrix is given by the Euler–Lotka equation. The Leslie model is very similar to a discrete-time Markov chain. The main difference is that in
Leslie_matrix
Vectors mapped to 0 by a linear map
N(A)=Null(A)=ker(A)={𝐱∈Kn∣A𝐱=𝟎}. The matrix equation is equivalent to a homogeneous system of linear equations: A𝐱=𝟎⇔a11x1+a12x2+⋯+a1nxn=0a21x1+a2
Kernel_(linear_algebra)
Quantum consistency equation
scattering matrix and if it satisfies the Yang–Baxter equation then the system is integrable. In quantum integrable models, the Yang–Baxter equation ensures
Yang–Baxter_equation
Differential equation that is linear with respect to the unknown function
In mathematics, a linear differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written
Linear_differential_equation
Matrix of geometric progressions
Exploiting the structure of the Vandermonde matrix, one can use Newton's divided differences method) to solve the equation in O(n2) time, which also gives the
Vandermonde_matrix
Matrix representing the effect of scattering on a physical system
scattering matrix – a unitary matrix of coefficients connecting "the asymptotic behaviour of an arbitrary particular solution [of the integral equations] with
S-matrix
Dimension of the column space of a matrix
the system of linear equations and linear transformation encoded by A. There are multiple equivalent definitions of rank. A matrix's rank is one of its
Rank_(linear_algebra)
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
Rotation_matrix
Equation used in quantum scattering problems
The Lippmann–Schwinger equation (named after Bernard Lippmann and Julian Schwinger) is one of the most used equations to describe elastic particle collisions –
Lippmann–Schwinger_equation
Matrix with non-zero elements only in a diagonal band
differential equation on a square domain (using central differences) will yield a matrix with a bandwidth equal to the square root of the matrix dimension
Band_matrix
System of equations in mathematics
differential-algebraic system of equations (DAE) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to
Differential-algebraic system of equations
Differential-algebraic_system_of_equations
Nonlinear second-order partial differential equation of special kind
Hessian matrix D2u, then it is an equation of Monge–Ampère type. As for other second-order fully nonlinear equations, the type of a Monge–Ampère equation is
Monge–Ampère_equation
Finite difference method for numerically solving parabolic differential equations
Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. It is a second-order
Crank–Nicolson_method
Dynamical system
prevalence relative to others. Another key difference from the quasispecies model is that the replicator equation does not include mechanisms for mutation
Replicator_equation
Mathematical method used in optics and acoustics
infinite number of reflections. The transfer-matrix method is based on the fact that, according to Maxwell's equations, there are simple continuity conditions
Transfer-matrix method (optics)
Transfer-matrix_method_(optics)
Finite difference equation
In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. In it, the discrete Laplace operator takes the place
Discrete_Poisson_equation
Branch of physics
methods involve: time-stepping through the equations over the whole domain for each time instant; banded matrix inversion to calculate the weights of basis
Computational electromagnetics
Computational_electromagnetics
Differential equation containing derivatives with respect to only one variable
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other
Ordinary differential equation
Ordinary_differential_equation
Differential equation exhibiting high rate of dissipation
stiff equation is the necessary condition Td(divuf)(u)≪−1. Since (divuf)(u)=trace(graduf)(u)=tracef′(u), where f′(u)∈ℝd×d is the Jacobian matrix of f at
Stiff_equation
Polynomial equation of degree 4
mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is ax4+bx3+cx2+dx+e=0
Quartic_equation
Kreiss to analyze the stability of finite difference methods for partial difference equations. Given a matrix A, the Kreiss constant 𝒦(A) (with respect
Kreiss_matrix_theorem
2005 mathematics textbook
roots of characteristic polynomials, the Leslie matrix in population dynamics, matrix difference equations and Markov chains, recurrences in modular arithmetic
Difference Equations: From Rabbits to Chaos
Difference_Equations:_From_Rabbits_to_Chaos
Assumption that motions of nuclei and electrons can be separated
the calculation of the nuclear Schrödinger equation. The BO approximation recognizes the large difference between the electron mass and the masses of
Born–Oppenheimer approximation
Born–Oppenheimer_approximation
Matrix representation of a graph
the negative continuous Laplacian obtained by the finite difference method. The Laplacian matrix relates to many functional graph properties. Kirchhoff's
Laplacian_matrix
Parameter in differential equations and dynamical systems
conditions is referred to as an initial value problem. A linear matrix difference equation of the homogeneous (having no constant term) form Xt+1=AXt has
Initial_condition
Statistical model to calculate the value of multiple quantities as they change over time
differenced d times and one has a VAR in difference. One can stack the vectors in order to write a VAR(p) as a stochastic matrix difference equation,
Vector_autoregression
Representation of a matrix as a product
efficient matrix algorithms. For example, when solving a system of linear equations A x = b {\displaystyle A\mathbf {x} =\mathbf {b} } , the matrix A can
Matrix_decomposition
Theorem in numerical analysis
finite difference methods for the numerical solution of linear partial differential equations. It states that for a linear consistent finite difference method
Lax_equivalence_theorem
Numerical solution method of computational electromagnetics
sometimes in acoustics, based on finite-difference approximations of the derivative operators in the differential equation being solved. While "FDFD" is a generic
Finite-difference frequency-domain method
Finite-difference_frequency-domain_method
Description of the time-evolution of plasma
In plasma physics, the Vlasov equation is a differential equation describing the time evolution of the distribution function of a collisionless plasma
Vlasov_equation
Equation that describes density changes of a material that is diffusing in a medium
definite matrix, and the equation is written (for three dimensional diffusion) as: ∂ϕ(𝐫,t)∂t=∑i=13∑j=13∂∂xi[Dij(ϕ,𝐫)∂ϕ(𝐫,t)∂xj] The diffusion equation has
Diffusion_equation
Antagonist in The Matrix (film series)
negative, a result of the Matrix's equation trying to balance itself. She tells Neo that Smith will destroy both the Matrix and the real world unless
Agent_Smith
Polynomial equation of degree 3
algebra, a cubic equation in one variable is an equation of the form ax3+bx2+cx+d=0 in which a is not zero. The solutions of this equation are called roots
Cubic_equation
Branch of ordinary differential equations
the linear differential equation form a vector space. A matrix ϕ ( t ) {\displaystyle \phi \,(t)} is called a fundamental matrix solution if the columns
Floquet_theory
Matrix equal to its conjugate-transpose
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex-valued entries that is equal to its own conjugate transpose
Hermitian_matrix
More equations than unknowns (mathematics)
the first one. Any system of linear equations can be written as a matrix equation. The previous system of equations (in Diagram #1) can be written as follows:
Overdetermined_system
Topics referred to by the same term
set up by a steady current density Young–Laplace equation, in physics, describing pressure difference over an interface in fluid mechanics Laplace expansion
Laplace's_law
branch of fundamental physics, the matrix representations of the Maxwell's equations are a formulation of Maxwell's equations using matrices, complex numbers
Matrix representation of Maxwell's equations
Matrix_representation_of_Maxwell's_equations
Scientific journal
Clarivate. Retrieved October 17, 2025. "Journal of Difference Equations and Applications". MIAR: Information Matrix for the Analysis of Journals. University of
Journal of Difference Equations and Applications
Journal_of_Difference_Equations_and_Applications
Mathematical model of a system in control engineering
system behavior over time through first-order differential equations or difference equations. These state variables change based on their current values
State-space_representation
electromagnetics, the scattering-matrix method (SMM) is a numerical method used to solve Maxwell's equations, related to the transfer-matrix method. SMM can, for
Scattering-matrix_method
In model checking, a field of computer science, a difference bound matrix (DBM) is a data structure used to represent some convex polytopes called zones
Difference_bound_matrix
Mathematical concept
linear equations (underdetermined or otherwise) is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If
Underdetermined_system
Field of mathematics
statistics. Matrix methods are particularly used in finite difference methods, finite element methods, and the modeling of differential equations. Noting
Numerical_linear_algebra
Class of second-order linear partial differential equations
A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent
Parabolic partial differential equation
Parabolic_partial_differential_equation
Method for solving continuous operator problems (such as differential equations)
previous equation is actually a linear system of equations Au=f, where Aij=a(ej,ei),fi=f(ei). Due to the definition of the matrix entries, the matrix of the
Galerkin_method
Concept in differential equation mathematics
the spectral radius of the update matrix A ( Δ t ) {\displaystyle A(\Delta t)} . For the linear structural equation M u ¨ + C u ˙ + K u = f ext {\displaystyle
Newmark-beta_method
Determinant of the matrix of first derivatives of a set of functions
(n−1)-st derivative, thus forming a square matrix. When the functions are solutions of a linear differential equation, the Wrońskian can be found explicitly
Wronskian
Polynomial function of degree 4
polynomial of the matrix. The characteristic equation of a fourth-order linear difference equation or differential equation is a quartic equation. An example
Quartic_function
Mathematical condition for convergence
partial differential operator has been approximated by a finite difference equation, which is then solved by numerical linear algebra methods. This quantity
Courant–Friedrichs–Lewy condition
Courant–Friedrichs–Lewy_condition
Numerical analysis technique
modeling computational electrodynamics. Finite difference schemes for time-dependent partial differential equations (PDEs) have been employed for many years
Finite-difference time-domain method
Finite-difference_time-domain_method
Methods used to find numerical solutions of ordinary differential equations
ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is
Numerical methods for ordinary differential equations
Numerical_methods_for_ordinary_differential_equations
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
Representation of a type of random process
the form of a stochastic difference equation (or recurrence relation) which should not be confused with a differential equation. Together with the moving-average
Autoregressive_model
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 model of the time dependence of a point in space
discrete-time, affine dynamical system has the form of a matrix difference equation: xn+1=Axn+b, with A a matrix and b a vector. As in the continuous case, the
Dynamical_system
Analog of the continuous Laplace operator
In matrix-vector notation, dϕdt=−k(D−A)ϕ=−kLϕ, which gives dϕdt+kLϕ=0. Notice that this equation takes the same form as the heat equation, where the
Discrete_Laplace_operator
Binary operation
its subgraphs. Redheffer, Raymond (1959). "Inequalities for a Matrix Riccati Equation". Journal of Mathematics and Mechanics. 8 (3): 349–367. ISSN 0095-9057
Redheffer_star_product
Numerical method in computational electromagnetics
expression into the equation yields: ∑nNanL(fn)≈g. We can also define a residual for this expression, which denotes the difference between the actual and
Method of moments (electromagnetics)
Method_of_moments_(electromagnetics)
Mathematical relationship describing the flow of groundwater through an aquifer
assumptions which went into both these equations are: the aquifer material is incompressible (no change in matrix due to changes in pressure — aka subsidence)
Groundwater_flow_equation
Relativistic wave equation in quantum mechanics
publish it. He lost confidence in the equation after he did not manage to use it to establish the equivalence between matrix mechanics and wave mechanics, like
Klein–Gordon_equation
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
Number with a real and an imaginary part
f(t) = ert. Likewise, in difference equations, the complex roots r of the characteristic equation of the difference equation system are used, to attempt
Complex_number
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
Equations describing classical electromagnetism
Maxwell's equations are a set of coupled partial differential equations that describe how electric and magnetic fields are generated by electric charges
Maxwell's_equations
Identity relating to differential equations
equation identity) is an equation that expresses the Wronskian of two solutions of a homogeneous second-order linear ordinary differential equation in
Abel's_identity
Numerical method for solving physical or engineering problems
method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas
Finite_element_method
Function's sensitivity to argument change
Practically, such a matrix is almost singular, and the computation of its inverse, or solution of a linear system of equations is prone to large numerical
Condition_number
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
Rewrite the equation using matrix transpose: I=m(𝐱𝖳𝐱−𝐧̂𝖳𝐱𝐱𝖳𝐧̂)=m⋅𝐧̂𝖳(𝐱𝖳𝐱⋅𝐄𝟑−𝐱𝐱𝖳)𝐧̂, where E3 is the 3×3 identity matrix. This leads
Moment_of_inertia
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MATRIX DIFFERENCE-EQUATION
MATRIX DIFFERENCE-EQUATION
MATRIX DIFFERENCE-EQUATION
MATRIX DIFFERENCE-EQUATION
MATRIX DIFFERENCE-EQUATION
MATRIX DIFFERENCE-EQUATION
MATRIX DIFFERENCE-EQUATION
MATRIX DIFFERENCE-EQUATION
MATRIX DIFFERENCE-EQUATION
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