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Generalization of signum function to matrices
In mathematics, the matrix sign function is a matrix function on square matrices analogous to the complex sign function. It was introduced by J.D. Roberts
Matrix_sign_function
Function returning minus 1, zero or plus 1
the sign function or signum function (from signum, Latin for "sign") is a function that has the value −1, +1 or 0 according to whether the sign of a
Sign_function
Function that maps matrices to matrices
In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of
Analytic_function_of_a_matrix
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
Matrix of second derivatives
the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar
Hessian_matrix
Orthogonal matrix
Walsh matrix is sequency-ordered. Confusingly, different sources refer to either matrix as the Walsh matrix. The Walsh matrix (and Walsh functions) are
Walsh_matrix
On converting relations to functions of several real variables
{0} \}.} To state the implicit function theorem, we need the Jacobian matrix of f {\displaystyle f} , which is the matrix of the partial derivatives of
Implicit_function_theorem
Sum of elements on the main diagonal
In linear algebra, the trace of a square matrix A, denoted tr(A), is defined as a sum of the elements on its main diagonal, a 11 + a 22 + ⋯ + a n n {\displaystyle
Trace_(linear_algebra)
In mathematics, invariant of square matrices
mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or
Determinant
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
Mathematical operation
square root of a Markov matrix: eigenvalues and the Taylor series Denman, Eugene D.; Beavers, Alex N. (1976), "The matrix sign function and computations in
Square_root_of_a_matrix
Fourier transform of the probability density function
probability density function, then the characteristic function is the Fourier transform (with sign reversal) of the probability density function. Thus it provides
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Technique in mathematical modeling
solution of matrix equations. The implementation is based on spectral projection methods, e.g., methods based on the matrix sign function and the matrix disk
Model_order_reduction
Specialized notation for multivariable calculus
derivative of a scalar function with respect to a matrix, known as the gradient matrix, which collects the derivative with respect to each matrix element in the
Matrix_calculus
Theorem in mathematics
replacing "derivative" with "Jacobian matrix" and "nonzero derivative" with "nonzero Jacobian determinant". If the function of the theorem belongs to a higher
Inverse_function_theorem
Array of numbers
set of vectors Irregular matrix Matrix calculus – Specialized notation for multivariable calculus Matrix function – Function that maps matrices to matricesPages
Matrix_(mathematics)
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
Matrix operation generalizing exponentiation of scalar numbers
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems
Matrix_exponential
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
Mathematical operation
On the graph of a function, the sign of the second derivative is related to the concavity of the graph. The graph of a function with a positive second
Second_derivative
Instantaneous rate of change (mathematics)
best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect
Derivative
Correlation of a signal with a time-shifted copy of itself, as a function of shift
autocorrelation matrix is a Hermitian matrix for complex random vectors and a symmetric matrix for real random vectors. The autocorrelation matrix is a positive
Autocorrelation
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
Polynomial of the elements of a matrix
In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant,
Permanent_(mathematics)
Multivariate derivative (mathematics)
denotes the transpose Jacobian matrix. For the second form of the chain rule, suppose that h : I → R is a real valued function on a subset I of R, and that
Gradient
Limit of a function One-sided limit Limit of a sequence Indeterminate form Orders of approximation (ε, δ)-definition of limit Continuous function Derivative
List_of_calculus_topics
Irreducible representation of the rotation group SO
\beta )} is the Bessel function and ℓ β {\displaystyle \ell \beta } is finite. Using sign convention of Wigner, et al. the d-matrix elements d m ′ m j (
Wigner_D-matrix
American filmmakers
Wayback Machine, The New York Times. Lana and Andy Wachowski New Matrix, Jupiter Ascending, Sign Vendetta Mask @ Cloud Atlas Gala London Archived May 3, 2017
The_Wachowskis
Conditions for switching order of integration in calculus
was used by Leonhard Euler. More formally, the theorem states that if a function is Lebesgue integrable on a rectangle X × Y {\displaystyle X\times Y}
Fubini's_theorem
Set of rules defining correctly structured programs
product intended for conventional matrix multiplication uses the + and × functions, replacing these with other dyadic functions can result in useful alternative
APL_syntax_and_symbols
Integer matrices with +1 or −1 determinant; invertible over the integers. GL_n(Z)
mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or −1. Equivalently, it is an integer matrix that is invertible over
Unimodular_matrix
Differential operator in mathematics
time-independent functions. The overall sign of the metric here is chosen such that the spatial parts of the operator admit a negative sign, which is the
Laplace_operator
Method of evaluating certain integrals along paths in the complex plane
meromorphic function is a pairing between a cohomology class of differential forms and a homology class of cycles in the domain of the function. It also
Contour_integration
Differential calculus on function spaces
J=J[y]-J[f]} has the same sign for all y {\displaystyle y} in an arbitrarily small neighborhood of f . {\displaystyle f.} The function f {\displaystyle f} is
Calculus_of_variations
Type of derivative in mathematics
a matrix-vector product. Other notations for the derivative include D a f {\displaystyle D_{a}f} and D f ( a ) {\displaystyle Df(a)} . A function is
Derivative (multivariable calculus)
Derivative_(multivariable_calculus)
Differentiation under the integral sign formula
δ → 0 may be passed through the integral sign. If instead we only know that there is an integrable function θ : Ω → R {\displaystyle \theta \colon \Omega
Leibniz_integral_rule
Point to which functions converge in analysis
mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which
Limit_of_a_function
Typographical symbol (@)
decorate a function (wrap the function in another one at creation time). In Python 3.5 and up, it is also used as an overloadable matrix multiplication
At_sign
Mathematical function having a characteristic S-shaped curve or sigmoid curve
unitary sigmoid functions of all three kinds. Table 1. Taxonomy matrix with examples of sigmoid functions of the 1st kind Table 2. Taxonomy matrix with examples
Sigmoid_function
Generalization of the one-dimensional normal distribution to higher dimensions
× k {\displaystyle k\times k} matrix Σ {\displaystyle {\boldsymbol {\Sigma }}} , such that the characteristic function of X {\displaystyle \mathbf {X}
Multivariate normal distribution
Multivariate_normal_distribution
Infinite sum
even for studying finite structures in combinatorics through generating functions. The mathematical properties of infinite series make them widely applicable
Series_(mathematics)
Polynomial whose roots are the eigenvalues of a matrix
algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It
Characteristic_polynomial
Mathematical function
{\displaystyle n\times n} matrix, and T {\displaystyle {}^{\mathsf {T}}} denotes matrix transposition. The integral of this Gaussian function over the whole n
Gaussian_function
For a square matrix, the transpose of the cofactor matrix
classical adjoint adj(A) of a square matrix A is the transpose of its cofactor matrix. It is occasionally known as adjunct matrix, or "adjoint", though that normally
Adjugate_matrix
Notion in statistics
some initial results by Francis Ysidro Edgeworth). The Fisher information matrix is used to calculate the covariance matrices associated with maximum-likelihood
Fisher_information
Formulation of quantum mechanics
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually
Matrix_mechanics
Theorem in vector calculus
affine linear function on R n {\displaystyle \mathbb {R} ^{n}} can then be written in the form A x + b {\displaystyle Ax+b} , where A is a matrix and b is
Stokes'_theorem
Theorem in mathematics
theorem) is a theorem about differentiable functions, roughly stating that the average rate of change of such a function over an interval is equal to the instantaneous
Mean_value_theorem
Type of financial fraud
A matrix scheme (also known as a matrix sale or site, and as a hellevator, excavator or ladder scheme) is a business model involving the exchange of money
Matrix_scheme
Mathematical symbols (+ and −)
The plus sign (+) and the minus sign (−) are mathematical symbols used to denote positive and negative functions, respectively. In addition, the symbol
Plus_and_minus_signs
Mathematical symbol
programming language to denote the sign function. The lower-case Latin letter x is sometimes used in place of the multiplication sign. This is considered incorrect
Multiplication_sign
Mathematical identities
identities involving derivatives and integrals in vector calculus. For a function f ( x , y , z ) {\displaystyle f(x,y,z)} in three-dimensional Cartesian
Vector_calculus_identities
Method of mathematical integration
of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The
Lebesgue_integral
Technique in integral evaluation
matrix of partial derivatives of φ at the point (u1, ..., un). This formula expresses the fact that the absolute value of the determinant of a matrix
Integration_by_substitution
Notation of differential calculus
derivative Hessian matrix – Matrix of second derivatives Jacobian matrix – Matrix of partial derivatives of a vector-valued functionPages displaying short
Notation_for_differentiation
Relationship between derivatives and integrals
differentiating a function (calculating its slopes, or rate of change at every point on its domain) with the concept of integrating a function (calculating
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Method for evaluating indefinite integrals
form of the function being integrated and on methods for integrating rational functions, radicals, logarithms, and exponential functions. Risch called
Risch_algorithm
Definite integral of a scalar or vector field along a path
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms path integral, curve integral
Line_integral
Branch of mathematical analysis
to iterative application of a linear operator D {\displaystyle D} to a function f {\displaystyle f} , that is, repeatedly composing D {\displaystyle D}
Fractional_calculus
Generalization of definite integrals to functions of multiple variables
by nested integral signs in the reverse order of execution (the leftmost integral sign is computed last), followed by the function and integrand arguments
Multiple_integral
Conjecture on zeros of the zeta function
corresponding random-matrix ensembles. For the Riemann zeta function, the relevant ensemble is that of the unitary group. Random matrix theory has also led
Riemann_hypothesis
Function related to statistics and probability theory
allow for differentiation under the integral sign. And lastly, it is assumed that the information matrix, I ( θ ) = ∫ − ∞ ∞ ∂ log f ∂ θ r ∂ log
Likelihood_function
Instantaneous rate of change of the function
the directional derivative measures the instantaneous rate at which a function changes along a specified vector through a given point. If the vector is
Directional_derivative
Statement relating differentiable symmetries to conserved quantities
action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of
Noether's_theorem
Circulation density in a vector field
maps Ck functions in R3 to Ck−1 functions in R3, and in particular, it maps continuously differentiable functions R3 → R3 to continuous functions R3 → R3
Curl_(mathematics)
Derivative of a function with multiple variables
function with respect to income. d'Alembert operator Chain rule Curl (mathematics) Divergence Exterior derivative Iterated integral Jacobian matrix and
Partial_derivative
Notion in calculus
calculus, the differential represents the principal part of the change in a function y = f ( x ) {\displaystyle y=f(x)} with respect to changes in the independent
Differential_of_a_function
Networking protocol for real-time communication
Matrix (sometimes stylized as [matrix] or [m] for short) is an open standard and communication protocol for real-time communication. It aims to make real-time
Matrix_(protocol)
Mathematical theorem
x_{i}}}\right).} In other words, the matrix of the second-order partial derivatives, known as the Hessian matrix, is a symmetric matrix. Sufficient conditions for
Symmetry of second derivatives
Symmetry_of_second_derivatives
complexity of matrix multiplication. 4. Written as a function of another function, it is used for comparing the asymptotic growth of two functions. See Big
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Certain vector fields are the sum of an irrotational and a solenoidal vector field
again mirrored at the diagonal, but with a negative sign. In the three-dimensional case, the matrix elements just correspond to the components of the vector
Helmholtz_decomposition
Mathematical approximation of a function
f(a) is the Hessian matrix. In order to compute a second-order Taylor series expansion around the point (a, b) = (0, 0) of the function f ( x , y ) = e x
Taylor_series
Expectation value of time-ordered quantum operators
correlation functions resulting in the same S-matrix and therefore describing the same physics. They are closely related to correlation functions between
Correlation function (quantum field theory)
Correlation_function_(quantum_field_theory)
Extension of evidence theory to continuous variables of interest
the density function does not exist. By using the fully swept moment matrix, we represent the vacuous linear belief functions as a zero matrix in the swept
Linear_belief_function
Formula in calculus
square root function and the function a + b z + c z 2 {\displaystyle a+bz+cz^{2}\!} . He first mentioned it in a 1676 memoir (with a sign error in the
Chain_rule
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
Covariance and correlation
Autocovariance Coherence Convolution Correlation Correlation function Cross-correlation matrix Cross-covariance Cross-spectrum Digital image correlation
Cross-correlation
Concept in probability theory and statistics
function, evaluated at 0. In addition to univariate real-valued distributions, moment generating functions can also be defined for vector- or matrix-valued
Moment_generating_function
Study of rates of change
analogous methods use the gradient and the Hessian matrix. A critical point of a scalar-valued function occurs where the gradient is zero, and the eigenvalues
Differential_calculus
Mathematical method in calculus
the signed measure corresponding to the function of bounded variation χ [ a , b ] ( x ) f ( x ) {\displaystyle \chi _{[a,b]}(x)f(x)} , and functions f ~
Integration_by_parts
Antiderivative of the secant function
In calculus, the integral of the secant function can be evaluated using a variety of methods and there are multiple ways of expressing the antiderivative
Integral of the secant function
Integral_of_the_secant_function
Point where the derivative of a function is zero or undefined (in certain cases)
Hessian matrix is nonsingular is said to be nondegenerate, and the signs of the eigenvalues of the Hessian determine the local behavior of the function. In
Critical_point_(mathematics)
\left|f(x)\right|\,dx=\operatorname {sgn}(f(x))g(x)+C,} where sgn(x) is the sign function, which takes the values −1, 0, 1 when x is respectively negative, zero
Lists_of_integrals
Mathematical operation in calculus
implicit differentiation is a method for finding the derivative of a function that is defined by an equation rather than by an explicit formula. If an
Implicit_differentiation
Indefinite integral
function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function
Antiderivative
Formula for the derivative of a product
formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as ( u ⋅ v ) ′ = u ′
Product_rule
Approximation of a function by a polynomial
theorem gives an approximation of a k {\textstyle k} -times differentiable function around a given point by a polynomial of degree k {\textstyle k} , called
Taylor's_theorem
Mathematical relation consisting of a multi-variable function equal to zero
multivariable functions that are continuously differentiable. A common type of implicit function is an inverse function. Not all functions have a unique
Implicit_function
Change of variable for integrals involving trigonometric functions
integrals, which converts a rational function of trigonometric functions of x {\textstyle x} into an ordinary rational function of t {\textstyle t} by setting
Tangent half-angle substitution
Tangent_half-angle_substitution
Vector operator in vector calculus
physical interpretation suggests. This is because the trace of the Jacobian matrix of an N-dimensional vector field F in N-dimensional space is invariant under
Divergence
Formula for the derivative of a ratio of functions
rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let h ( x ) = f ( x ) g ( x ) {\displaystyle
Quotient_rule
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
Theorem in calculus
through a source or sink (with the flow through a sink given a negative sign) is equal to the divergence of the velocity field at the pipe mouth, so adding
Divergence_theorem
Matrix with exactly one 1 per row and column
In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column
Permutation_matrix
Mathematical description of quantum state
permanent (of a matrix) was part of the method, provided by John C. Slater. Schrödinger did encounter an equation for the wave function that satisfied
Wave_function
information matrix — a matrix representing the variance of the partial derivative, with respect to a parameter, of the log of the likelihood function of a random
List_of_named_matrices
in a more general context which was concerned with integrable matrix-valued functions that also had integrable log determinant. Because applications
Polynomial matrix spectral factorization
Polynomial_matrix_spectral_factorization
Technique of integral evaluation
In mathematics, a trigonometric substitution replaces a trigonometric function for another expression. In calculus, trigonometric substitutions are a technique
Trigonometric_substitution
Vector calculus formulas relating the bulk with the boundary of a region
matrix}\mathbf {P} ({\boldsymbol {\eta }})&{\text{for }}~{\boldsymbol {\eta }}\in U\\0&{\text{for }}~{\boldsymbol {\eta }}\not \in U\\\end{matrix}}\right
Green's_identities
Arithmetic operation
square matrix, then the product of A with itself n times is called the matrix power. Also A 0 {\displaystyle A^{0}} is defined to be the identity matrix, and
Exponentiation
MATRIX SIGN-FUNCTION
MATRIX SIGN-FUNCTION
Female
Finnish
Finnish form of Greek Margarites, MAARIT means "pearl."
Female
Welsh
Welsh form of Old French Caterine, CATRIN means "pure."
Female
German
Pet form of German Sieglinde, SIGI means "gentle battle."Â Compare with masculine Sigi.
Female
English
French form of Latin Maria, MARIE means "obstinacy, rebelliousness" or "their rebellion."
Girl/Female
Australian, Danish, Finnish, German, Latin, Scandinavian, Swedish
Sign; New Victory
Female
English
Pet form of English Matilda, MATTIE means "mighty in battle." Compare with masculine Mattie.
Male
English
Pet form of English Martin, MARTIE means "of/like Mars."
Male
English
Pet form of English Matthew, MATTIE means "gift of God." Compare with feminine Mattie.
Girl/Female
Arabic, Australian, Basque, French, Latin
Lady; Feminine of Martin; Warlike
Male
German
Pet form of Old High German Siegfried, SIGI means "victory-peace." Compare with feminine Sigi.Â
Female
Norse
Variant spelling of Old Norse Signy, SIGNE means "new victory."
Girl/Female
Latin American Swedish
Sign.
Female
German
Pet form of German Katarine, KATRIN means "pure."
Female
English
English form of Latin Viatrix, BEATRIX means "voyager (through life)."
Male
English
 English form of Roman Latin Martinus, MARTIN means "of/like Mars." Compare with another form of Martin.
Girl/Female
Danish, German, Latin, Scandinavian, Swedish
Sign; Signal; Victory
Girl/Female
Norse
Daughter of Volsung.
Male
French
 French form of Roman Latin Martinus, MARTIN means "of/like Mars." Compare with another form of Martin.
Girl/Female
Latin
Sign.
Female
Finnish
Finnish form of Greek Maria, MAARIA means "obstinacy, rebelliousness" or "their rebellion."Â
MATRIX SIGN-FUNCTION
MATRIX SIGN-FUNCTION
Girl/Female
American, Arabic, Australian, Danish, Swedish
Night; Born at Night; Sweetheart; Dark-haired Beauty
Female
English
Variant form of English Rachel, RACHYL means "ewe."
Boy/Male
Afghan, Arabic, German, Indian, Iranian, Muslim, Parsi, Persian
Agreement; Will; Resignation; Contentment; Acceptance; Satisfaction
Girl/Female
American, Christian, Hebrew, Indian, Swedish, Telugu
Avenged or Judged and Vindicated; Justified; Decoration; Valley; Spear Ruler
Boy/Male
Hindu, Indian, Telugu
One Part from Vedas
Boy/Male
Scottish
Son of the conquering people.
Boy/Male
Australian, Bengali, German, Hebrew, Indian
A Farmer; Bringer of Lighto; Shining; Giving Light; Glowing
Girl/Female
Tamil
Javaahari | ஜவாஹரீÂ
Boy/Male
Indian, Punjabi, Sikh
Unique King
Biblical
foxes; fists; path
MATRIX SIGN-FUNCTION
MATRIX SIGN-FUNCTION
MATRIX SIGN-FUNCTION
MATRIX SIGN-FUNCTION
MATRIX SIGN-FUNCTION
n.
Hence, one of the gestures of pantomime, or of a language of a signs such as those used by the North American Indians, or those used by the deaf and dumb.
a.
Having the negative sign, or sign minus.
v. i.
To make a sign or signal; to communicate directions or intelligence by signs.
n.
An embodiment of sin; a very wicked person.
a.
Of or pertaining to the Maoris or to their language.
n.
A word or a character regarded as the outward manifestation of thought; as, words are the sign of ideas.
a.
Of or pertaining to the meter as a standard of measurement; of or pertaining to the decimal system of measurement of which a meter is the unit; as, the metric system; a metric measurement.
v. t.
To influence by singing; to lull by singing; as, to sing a child to sleep.
n.
To make a sign upon; to mark with a sign.
pl.
of Matrix
n.
A sin offering; a sacrifice for sin.
n.
See Matrix.
p. pr. & vb. n.
of Sign
pl.
of Maori
n.
A character indicating the relation of quantities, or an operation performed upon them; as, the sign + (plus); the sign -- (minus); the sign of division Ö, and the like.
v. t.
To affix one's signature to, a second time; to sign again.
n.
To represent by a sign; to make known in a typical or emblematic manner, in distinction from speech; to signify.
v. i.
To be a sign or omen.
n.
A mold; a matrix.