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Matrix with one nonzero entry in each row and column
mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly
Generalized permutation matrix
Generalized_permutation_matrix
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
sum of permutations Enumerations of specific permutation classes Factorial Falling factorial Permutation matrix Generalized permutation matrix Inversion
List_of_permutation_topics
Array of numbers
can be described in matrix form, with a mass matrix multiplying a generalized velocity to give the kinetic term, and a force matrix multiplying a displacement
Matrix_(mathematics)
matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries
List_of_named_matrices
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
In mathematics, invariant of square matrices
Generalizing the above to higher dimensions, the determinant of an n × n {\displaystyle n\times n} matrix is an expression involving permutations and
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
Antisymmetric permutation object acting on tensors
epsilon represents a collection of numbers defined from the sign of a permutation of the natural numbers 1, 2, ..., n, for some positive integer n. It
Levi-Civita_symbol
Statistics models class
In statistics, a generalized additive model (GAM) is a generalized linear model in which the linear response variable depends linearly on unknown smooth
Generalized_additive_model
Real square matrix whose columns and rows are orthogonal unit vectors
reflection matrix with θ = 90° generates a reflection about the line at 45° given by y = x and therefore exchanges x and y; it is a permutation matrix, with
Orthogonal_matrix
Norm on a vector space of matrices
{\displaystyle \phi (Px)=\phi (x)} for any permutation matrix P {\displaystyle P} . A norm is a unitarily invariant matrix norm if and only if it is a symmetric
Matrix_norm
Machine learning technique
n}} be permutation matrices; and D ∈ R m × n {\displaystyle D\in \mathbb {R} ^{m\times n}} an arbitrary matrix. The softmax function is permutation equivariant
Attention_(machine_learning)
Mathematical function generalizing the determinant and permanent
multilinear in the rows and columns of the matrix; and the immanant is invariant under simultaneous permutations of the rows or columns by the same element
Immanant
In mathematics, a generalized Clifford algebra (GCA) is a unital associative algebra that generalizes the Clifford algebra, and goes back to the work
Generalized_Clifford_algebra
Matrix representation of a graph
theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix, or discrete Laplacian, is a matrix representation of a
Laplacian_matrix
Exact statistical hypothesis test
A permutation test (also called re-randomization test or shuffle test) is an exact statistical hypothesis test. A permutation test involves two or more
Permutation_test
Class of statistical models
In statistics, a generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing
Generalized_linear_model
Families of matrices in mathematics, physics, and quantum information
particular quantum information, the term generalized Pauli matrices refers to families of matrices which generalize the (linear algebraic) properties of the
Generalizations of Pauli matrices
Generalizations_of_Pauli_matrices
Concept in combinatorics
The statistics of random permutations, such as the cycle structure of a random permutation, are of fundamental importance in the analysis of algorithms
Random_permutation_statistics
Mathematical function
}+\left[{\begin{matrix}n-1\\k-1\end{matrix}}\right]_{\alpha }+\delta _{n,0}\delta _{k,0}\,.} These generalized α-factorial coefficients then generate
Double_factorial
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
Polynomial of the elements of a matrix
set of all m-permutations of the n-set {1,2,...,n}. Ryser's computational result for permanents also generalizes. If A is an m × n matrix with m ≤ n, let
Permanent_(mathematics)
Name of two different techniques based on the singular value decomposition
a single-matrix SVD. The generalized singular value decomposition (GSVD) is a matrix decomposition on a pair of matrices which generalizes the singular
Generalized singular value decomposition
Generalized_singular_value_decomposition
Mathematical model
context. The permutation matrices are precisely the alternating sign matrices that don't contain −1. An example of an alternating sign matrix that is not
Alternating_sign_matrix
Concepts from linear algebra
diagonalizable. A matrix that is not diagonalizable is said to be defective. For defective matrices, the notion of eigenvectors generalizes to generalized eigenvectors
Eigenvalues_and_eigenvectors
Statistical test
matrices are subjected to random permutations many times, with the correlation being recalculated after each permutation. The significance of the observed
Mantel_test
Count of permutations by cycles
kind arise in the study of permutations. In particular, the unsigned Stirling numbers of the first kind count permutations according to their number of
Stirling numbers of the first kind
Stirling_numbers_of_the_first_kind
Group of symmetries of an n-dimensional hypercube
symmetries of the hypercube, the even-signed permutation group (the Coxeter group of type D), and the generalized alternating group. The alternating subgroup
Hyperoctahedral_group
Property in group theory
method for determining whether a given permutation is even or odd is to construct the corresponding permutation matrix and compute its determinant. The value
Parity_of_a_permutation
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
Type of matrix factorization
decomposition). The product sometimes includes a permutation matrix as well. LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers
LU_decomposition
Algorithms for matrix decomposition
least if B is a non-negative monomial matrix. In this simple case it will just correspond to a scaling and a permutation. More control over the non-uniqueness
Non-negative matrix factorization
Non-negative_matrix_factorization
Matrix of binary truth values
matrix, binary matrix, relation matrix, Boolean matrix, or (0, 1)-matrix is a matrix with entries from the Boolean domain B = {0, 1}. Such a matrix can
Logical_matrix
Mathematical operation on vectors in 3D space
to generalize the cross product to higher dimensions. The cross product can be seen as one of the simplest Lie products, and is thus generalized by Lie
Cross_product
Representation of a matrix as a product
triangular, and P is a permutation matrix. Existence: An LUP decomposition exists for any square matrix A. When P is an identity matrix, the LUP decomposition
Matrix_decomposition
Concept in statistics
statistics, the class of vector generalized linear models (VGLMs) was proposed to enlarge the scope of models catered for by generalized linear models (GLMs). In
Vector generalized linear model
Vector_generalized_linear_model
Number line and triangular tiling's symmetry mathematical structure
certain inequalities. Their procedure uses the matrix representation of affine permutations and generalizes the shadow construction, introduced in (Viennot
Affine_symmetric_group
Concept in mathematics
semistandard tableau for Q as well. The two-line array (or generalized permutation) wA corresponding to a matrix A is defined as w A = ( i 1 i 2 … i m j 1 j 2 …
Robinson–Schensted–Knuth correspondence
Robinson–Schensted–Knuth_correspondence
Special kind of square matrix
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal
Triangular_matrix
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
Tool for working with matrices
algorithm) is an algorithm for decomposing a bistochastic matrix into a convex combination of permutation matrices. It was published by Garrett Birkhoff in 1946
Birkhoff_algorithm
Problem in linear algebra
naively expands the formula, summing over all permutations and within the sum multiplying out each matrix entry. This requires n! n arithmetic operations
Computing_the_permanent
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
{\displaystyle 1/p!} in § Properties of the generalized Kronecker delta below disappearing. In terms of the indices, the generalized Kronecker delta is defined as:
Kronecker_delta
Mathematical operation on matrices
a zero matrix, and k is a scalar. Non-commutative: In general, A ⊗ B and B ⊗ A are different matrices. However, A ⊗ B and B ⊗ A are permutation equivalent
Kronecker_product
Algorithm for modelling sequential data
PM_{\text{causal}}P^{-1}} , where P {\displaystyle P} is a random permutation matrix. An encoder consists of an embedding layer, followed by multiple encoder
Transformer_(deep_learning)
Sum of elements on the main diagonal
The trace of a Hermitian matrix is real, because the elements on the diagonal are real. The trace of a permutation matrix is the number of fixed points
Trace_(linear_algebra)
Type of group in abstract algebra
can be generalized to the symmetric group of any finite totally ordered set, but not to that of an unordered set). The order reversing permutation is the
Symmetric_group
Measure of linear correlation
are a permutation of the set {1,...,n}. The permutation i′ is selected randomly, with equal probabilities placed on all n! possible permutations. This
Pearson correlation coefficient
Pearson_correlation_coefficient
Generalizability theory Generalized additive model Generalized additive model for location, scale and shape Generalized beta distribution Generalized
List_of_statistics_articles
Branch of mathematics
where Sn is the group of all permutations of n elements, σ is a permutation, and (−1)σ the parity of the permutation. A matrix is invertible if and only
Linear_algebra
Mathematical ring whose elements are matrices
abstract algebra, a matrix ring is a set of matrices with entries in a ring R that form a ring under matrix addition and matrix multiplication. The set
Matrix_ring
Block diagonal matrix of Jordan blocks
the mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring R (whose identities
Jordan_matrix
Statistical linear model
McCullagh, P.; Nelder, J. A. (January 1, 1983). "An outline of generalized linear models". Generalized Linear Models. Springer US. pp. 21–47. doi:10.1007/978-1-4899-3242-6_2
General_linear_model
Array representation in computer memory
the rows and columns of a two-dimensional array, i.e. a matrix, the orders can be generalized to arrays of any dimension by noting that the terms row-major
Row-_and_column-major_order
Partition into subsets from a given family
on to explain that it is better working with the generalized problem directly, because the generalized algorithm is simpler and faster: A simple change
Exact_cover
Operation on mathematical functions
a permutation group); and one says that the group is generated by these functions. The set of all bijective functions f: X → X (called permutations) forms
Function_composition
Generalization of the one-dimensional normal distribution to higher dimensions
{Q_{2}} } is a matrix, q 1 {\displaystyle {\boldsymbol {q_{1}}}} is a vector, and q 0 {\displaystyle q_{0}} is a scalar), is a generalized chi-squared variable
Multivariate normal distribution
Multivariate_normal_distribution
Method of data analysis
the data's covariance matrix. Thus, the principal components are often computed by eigendecomposition of the data covariance matrix or singular value decomposition
Principal_component_analysis
Large language model developed by Google AI
&\ddots &\vdots \\0&0&0&\dots &0\end{bmatrix}}} permuted by a random permutation matrix to P M causal P − 1 {\displaystyle PM_{\text{causal}}P^{-1}} . The
XLNet
queuing systems The inverse-gamma distribution The generalized gamma distribution The generalized Pareto distribution The Gamma/Gompertz distribution
List of probability distributions
List_of_probability_distributions
Generalization of a magic square
n-1)] : component permutation ^[perm(0..n-1)] : coordinate permutation (n == 2: transpose) _2axis[perm(0..m-1)] : monagonal permutation (axis ε [0..n-1])
Magic_hypercube
Generalization of graph theory
some general directed graph. The generalized incidence matrix for such hypergraphs is, by definition, a square matrix, of a rank equal to the total number
Hypergraph
Polynomial with all terms of degree two
associated symmetric matrix is diagonal. Moreover, the coefficients λ1, λ2, ..., λn are determined uniquely up to a permutation. If the change of variables
Quadratic_form
A matrix canonical form
closed field is similar to a Weyr matrix W {\displaystyle W} which is unique up to permutation of its basic blocks. The matrix W {\displaystyle W} is called
Weyr_canonical_form
Function returning minus 1, zero or plus 1
{\displaystyle (\operatorname {sgn} 0)^{2}=0} . This generalized signum allows construction of the algebra of generalized functions, but the price of such generalization
Sign_function
Mathematical problems related to differential equations
Hilbert, and modern matrix-valued Riemann–Hilbert problems play a central role in integrable systems, orthogonal polynomials, random matrix theory, inverse
Riemann–Hilbert_problem
Classification of semi-simple rings and algebras
finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely determined up to permutation of the index i. In
Wedderburn–Artin_theorem
Mathematical problem
and permutations of rows and columns. A {1, −1} matrix is normalized if all elements in its first row and column equal 1. When the size of a matrix is
Hadamard's maximal determinant problem
Hadamard's_maximal_determinant_problem
Probabilistic problem-solving algorithm
approximate randomization and permutation tests. An approximate randomization test is based on a specified subset of all permutations (which entails potentially
Monte_Carlo_method
\end{aligned}}} Compositions of matrix inverse along the main diagonal, y ^ 1 {\displaystyle {\hat {y}}_{1}} and a generalized inverse along the secondary
Partial_inverse_of_a_matrix
Statistical property
based on the assumption of homoskedasticity is misleading. In that case, generalized least squares (GLS) was frequently used in the past. Nowadays, standard
Homoscedasticity and heteroscedasticity
Homoscedasticity_and_heteroscedasticity
Architectural motif in neural networks for aggregating information
of the whole graph. The global pooling layer must be permutation invariant, such that permutations in the ordering of graph nodes and edges do not alter
Pooling_layer
Type of mathematical group
representation over K). Any finite group is linear, because it can be realized by permutation matrices using Cayley's theorem. Among infinite groups, linear groups
Linear_group
Matrix decomposition
beginning of each new step—column pivoting— and thus introduces a permutation matrix P: A P = Q R ⟺ A = Q R P T {\displaystyle AP=QR\quad \iff \quad A=QRP^{\textsf
QR_decomposition
Mathematics optimization problem
Matrix chain multiplication (or the matrix chain ordering problem) is an optimization problem concerning the most efficient way to multiply a given sequence
Matrix_chain_multiplication
Matrix decomposition method
Formally, if A is an n × n positive semidefinite matrix of rank r, then there is at least one permutation matrix P such that P A PT has a unique decomposition
Cholesky_decomposition
Rational number sequence
alternating permutations of odd size are enumerated by the Euler numbers of odd index (also called tangent numbers) and the alternating permutations of even
Bernoulli_number
Concept in mathematics
In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some
Frobenius_group
Statistics concept
_{m}x_{i}^{m}+\varepsilon _{i}\ (i=1,2,\dots ,n)} can be expressed in matrix form in terms of a design matrix X {\displaystyle \mathbf {X} } , a response vector y →
Polynomial_regression
Statistical measure of association for two binary variables
observations. While there is no perfect way of describing the confusion matrix of true and false positives and negatives by a single number, the Matthews
Phi_coefficient
Statistical relationship
less structured relationships between variables. The concept has been generalized to other forms of association between two variables, such as mutual information
Correlation
Square root of the determinant of a skew-symmetric square matrix
determinant of an m-by-m skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries, a polynomial with integer coefficients
Pfaffian
Graph with tight clique-coloring relation
the scheduled time of each task. Both interval graphs and permutation graphs are generalized by the trapezoid graphs. Systems of intervals in which no
Perfect_graph
Tree-based ensemble machine learning methods
estimate of the generalization error. Measuring variable importance through permutation. The report also offers the first theoretical result for random forests
Random_forest
Natural number
5)+(6)\end{aligned}}} The product between neighbor numbers of 23, the dual permutation of the digits of 32 in decimal, is equal to the sum of the first 32 integers:
32_(number)
Statistical modeling method
a matrix B replacing the vector β of the classical linear regression model. Multivariate analogues of ordinary least squares (OLS) and generalized least
Linear_regression
Blockmodeling process
(CONCOR and STRUCTURE) that were used to "find a permutation of the rows and columns in the adjacency matrix leading to an approximate block structure". The
Deterministic_blockmodeling
Type of group in mathematics
than 1), and in an orthogonal matrix these entries must be in different coordinates, which is exactly the signed permutation matrices. In odd dimension,
Orthogonal_group
Time series model
average (ARMA) model is assumed for the error variance, the model is a generalized autoregressive conditional heteroskedasticity (GARCH) model. ARCH models
Autoregressive conditional heteroskedasticity
Autoregressive_conditional_heteroskedasticity
Group that admits a formal description in terms of reflections
Coxeter matrix is the n × n {\displaystyle n\times n} symmetric matrix with entries m i j {\displaystyle m_{ij}} . Indeed, every symmetric matrix with diagonal
Coxeter_group
Recursive integer sequence
illustrate the case n = 4: Cn is the number of stack-sortable permutations of {1, ..., n}. A permutation w is called stack-sortable if S(w) = (1, ..., n), where
Catalan_number
Function in discrete mathematics
convolution into pointwise product is the DFT up to a permutation of coefficients. Since the number of permutations of n elements equals n!, there exist exactly
Discrete_Fourier_transform
consistent with respect to permutations of the rows or columns of G {\displaystyle \mathrm {G} } . The RGA is often generalized in practice to be used when
Relative_gain_array
Number of values in the final calculation of a statistic that are free to vary
Robert Tibshirani (1990), Generalized additive models, CRC Press, (p. 54) and (eq.(B.1), p. 305)) Simon N. Wood (2006), Generalized additive models: an introduction
Degrees of freedom (statistics)
Degrees_of_freedom_(statistics)
Fast Fourier Transform algorithm
(The results are in the correct order in X and no further bit-reversal permutation is required; the often-mentioned necessity of a separate bit-reversal
Cooley–Tukey_FFT_algorithm
Statistical model for a binary dependent variable
the proposed model to every permutation of the yk and it can be shown that the maximum log-likelihood of these permutation fits will never be smaller than
Logistic_regression
Concept in probability theory and statistics
{\hat {\Sigma }}} is the sample covariance matrix, T {\displaystyle T} is a target matrix (e.g., a diagonal matrix), and the shrinkage intensity λ ∈ ( 0
Partial_correlation
Determinant of a product of rectangular matrices
the permutation matrix for π, ( R g ) S , [ m ] {\displaystyle (R_{g})_{S,[m]}} is the permutation matrix for σ, and LfRg is the permutation matrix for
Cauchy–Binet_formula
Surjective bounded operator on a Hilbert space preserving the inner product
basis is unitary. In the finite dimensional case, such operators are the permutation matrices. On the vector space C of complex numbers, multiplication by
Unitary_operator
GENERALIZED PERMUTATION-MATRIX
GENERALIZED PERMUTATION-MATRIX
Girl/Female
Hindu
Achievement, Omnipresence, Permeation
Boy/Male
Hindu, Indian, Jain, Marathi, Sanskrit, Sindhi, Tamil
Lines on Any Particular Raaga from Sanskrit; Permutations and Combinations of Parents; Aarya Cost King Ashoka's Birth
Girl/Female
Tamil
Vyaapti | வà¯à®¯à®¾à®ªà®¤à¯€
Achievement, Omnipresence, Permeation
Vyaapti | வà¯à®¯à®¾à®ªà®¤à¯€
Surname or Lastname
English
English : status name from Middle English squyer ‘esquire’, ‘a man belonging to the feudal rank immediately below that of knight’ (from Old French esquier ‘shield bearer’). At first it denoted a young man of good birth attendant on a knight, or by extension any attendant or servant, but by the 14th century the meaning had been generalized, and referred to social status rather than age. By the 17th century, the term denoted any member of the landed gentry, but this is unlikely to have influenced the development of the surname.
GENERALIZED PERMUTATION-MATRIX
GENERALIZED PERMUTATION-MATRIX
Boy/Male
English American French
Spear strength.
Girl/Female
Assamese, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi
Good Looking
Surname or Lastname
English (Lancashire)
English (Lancashire) : habitational name from Shorrock Green in Lancashire, probably so named from Old English scora ‘bank’ + Äc ‘oak’.
Boy/Male
Scottish Irish
From the craggy hills.' Tor is a name for a craggy hilltop and also may refer to a watchtower.
Boy/Male
Hindu, Indian, Punjabi, Rajasthani, Sikh
Powerful
Boy/Male
German Teutonic
Advancing wolf.
Surname or Lastname
English
English : unexplained.
Girl/Female
Hindu, Indian
Silk
Girl/Female
Norse
Goddess of skiers.
Boy/Male
African, American, Australian, British, English, Irish, Jamaican
Fair-haired; Based on a Surname and Place Name; Occasionally Used as a First Name
GENERALIZED PERMUTATION-MATRIX
GENERALIZED PERMUTATION-MATRIX
GENERALIZED PERMUTATION-MATRIX
GENERALIZED PERMUTATION-MATRIX
GENERALIZED PERMUTATION-MATRIX
v. t.
To derive or deduce (a general conception, or a general principle) from particulars.
v. i.
To form into a genus; to view objects in their relations to a genus or class; to take general or comprehensive views.
v. t.
To impregnate with a mineral; as, mineralized water.
n.
Barter; exchange.
a.
Comprising structural characters which are separated in more specialized forms; synthetic; as, a generalized type.
n.
Alt. of Perduration
n.
One who takes general or comprehensive views.
n.
Permutation.
n.
The arrangement of any determinate number of things, as units, objects, letters, etc., in all possible orders, one after the other; -- called also alternation. Cf. Combination, n., 4.
v. t.
Alteration in the order of a series; permutation.
n.
A generalized concept of magnitude.
n.
Long continuance.
n.
The act of drinking excessively; a drinking bout.
p. pr. & vb. n.
of Generalize
n.
The act of permuting; exchange of the thing for another; mutual transference; interchange.
n.
Any one of such possible arrangements.
v. t.
To bring under a genus or under genera; to view in relation to a genus or to genera.
v. t.
To apply to other genera or classes; to use with a more extensive application; to extend so as to include all special cases; to make universal in application, as a formula or rule.
v. t.
To make universal; to generalize.
imp. & p. p.
of Generalize