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Matrix-valued random variable
mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled randomly from a probability
Random_matrix
Measure of covariance of components of a random vector
covariance between each pair of elements of a given random vector. Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions
Covariance_matrix
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
Riemann_hypothesis
Generalization of gamma distribution to multiple dimensions
the distribution in 1928. Other names include Wishart ensemble (in random matrix theory, probability distributions over matrices are usually called "ensembles")
Wishart_distribution
Random variable with multiple component dimensions
of aggregate random variables, e.g. a random matrix, random tree, random sequence, stochastic process, etc. Formally, a multivariate random variable is
Multivariate_random_variable
Array of numbers
In mathematics, a matrix (pl.: matrices) is a rectangular array of numbers or other mathematical objects with elements or entries arranged in rows and
Matrix_(mathematics)
Technique to reduce dimensionality of points in Euclidean space
subspace. Random projection is computationally simple: form the random matrix "R" and project the d × N {\displaystyle d\times N} data matrix X onto K
Random_projection
an N×N Euclidean random matrix  is defined with the help of an arbitrary deterministic function f(r, r′) and of N points {ri} randomly distributed in a
Euclidean_random_matrix
Stochastic process
named for Freeman Dyson. Dyson studied this process in the context of random matrix theory. There are several equivalent definitions: Definition by stochastic
Dyson_Brownian_motion
British mathematician
is a British mathematician at the University of Bristol working in random matrix theory and quantum chaos. Snaith was educated at the University of Bristol
Nina_Snaith
functions of a matrix argument have applications in random matrix theory. For example, the distributions of the extreme eigenvalues of random matrices are
Hypergeometric function of a matrix argument
Hypergeometric_function_of_a_matrix_argument
Technologist
statistics, information and communication sciences with a special focus on random matrix theory and learning algorithms. In the AI field, he is known for his
Mérouane_Debbah
Family of probability distributions
The ranked eigenvalues En from these random matrices obey Wigner's semicircular distribution: For an N × N matrix the average density for eigenvalues of
Tweedie_distribution
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
Probability distribution
random matrix theory introduced by Craig Tracy and Harold Widom (1993, 1994). It is the distribution of the normalized largest eigenvalue of a random
Tracy–Widom_distribution
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
Type of matrix in probability theory and statistics
a cross-covariance matrix is a matrix whose element in the i, j position is the covariance between the i-th element of a random vector and j-th element
Cross-covariance_matrix
Probability distribution
distribution to matrix-valued random variables. The probability density function for the random matrix X (n × p) that follows the matrix normal distribution
Matrix_normal_distribution
e.g. a random matrix, random tree, random sequence, random process, etc. A random matrix is a matrix-valued random element. Many important properties
Random_element
Random matrix with gaussian entries
In random matrix theory, the Gaussian ensembles are specific probability distributions over self-adjoint matrices whose entries are independently sampled
Gaussian_ensemble
Concept in quantum mathematics
and measurements of a quantum circuit. The idea is similar to that of random matrix theory which is to use the QRC to obtain almost exact results of non-integrable
Quantum_random_circuits
Randomized mathematical sequence based upon the Fibonacci sequence
general class of random matrix products, the norm grows as λn, where n is the number of factors. Their results apply to a broad class of random sequence generating
Random_Fibonacci_sequence
Distribution of singular values of large rectangular random matrices
denotes an m × n {\displaystyle m\times n} random matrix whose entries are independent identically distributed random variables with mean 0 and variance σ 2
Marchenko–Pastur_distribution
When an ecosystem does not drastically change over time even after perturbation
are positive. The matrix J {\displaystyle J} is also known as the community matrix. May supposed that the Jacobian was a random matrix whose off-diagonal
Ecological_stability
and uniformly bounded random variables. In the matrix setting, the analogous theorem concerns a sum of positive-semidefinite random matrices subjected to
Matrix_Chernoff_bound
Australian and American mathematician (born 1975)
study of non-symmetric random matrices. They showed that if n is large and the entries of a n × n matrix A are selected randomly according to any fixed
Terence_Tao
Mathematical theory on random variables
generated free groups have the same elementary theory. Later connections to random matrix theory, combinatorics, representations of symmetric groups, large deviations
Free_probability
American mathematician
numerical linear algebra, computational science, parallel computing, and random matrix theory. He is one of the creators of the technical programming language
Alan_Edelman
Matrix with non-zero elements only in a diagonal band
In mathematics, particularly matrix theory, a band matrix or banded matrix is a sparse matrix whose non-zero entries are confined to a diagonal band, comprising
Band_matrix
Matrix of geometric progressions
In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row:
Vandermonde_matrix
Branch of physics seeking to explain chaotic dynamical systems in terms of quantum theory
born out of a desire to quantify spectral features of complex systems. Random matrix theory was developed in an attempt to characterize spectra of complex
Quantum_chaos
Generalization of the one-dimensional normal distribution to higher dimensions
covariance matrix is called the precision matrix, denoted by Q = Σ − 1 {\displaystyle {\boldsymbol {Q}}={\boldsymbol {\Sigma }}^{-1}} . A real random vector
Multivariate normal distribution
Multivariate_normal_distribution
Indian physicist (1932–2006)
an Indian theoretical physicist, particularly known for his work in random matrix theory. Madan Lal Mehta was born on 24 December 1932 in Relmagra, Rajasthan
Madan_Lal_Mehta
Discrete probability distribution
}})^{2},\alpha (1+{\sqrt {\lambda }})^{2}].} This law also arises in random matrix theory as the Marchenko–Pastur law. Its free cumulants are equal to
Poisson_distribution
Multivariable generalization of the Student's t-distribution
random vectors of the Student's t-distribution, which is a distribution applicable to univariate random variables. While the case of a random matrix could
Multivariate_t-distribution
American mathematician (born 1977)
known for work on sparse approximation, numerical linear algebra, and random matrix theory. Tropp studied at the University of Texas, where he completed
Joel_Tropp
Mathematical conjecture
the distribution would agree with the distribution of spacings of GUE random matrix eigenvalues using Cray X-MP. In 1987 he reported the calculations in
Montgomery's pair correlation conjecture
Montgomery's_pair_correlation_conjecture
South African mathematician
mathematician known for his work on spectral theory, integrable systems, random matrix theory and Riemann–Hilbert problems. Deift was born in Durban, South
Percy_Deift
1999 film by the Wachowskis
The Matrix is a 1999 science fiction action film written and directed by the Wachowskis. The first installment in the Matrix film series, it stars Keanu
The_Matrix
American physicist
a fellow of the American Physical Society, "for the development of random-matrix theory methods and their applications in atomic nuclei and in nonperturbative
Jacobus_Verbaarschot
Concept in information theory
the participation ratio, widely used in condensed matter physics and random matrix theory to measure the effective number of states contributing to a distribution
Rényi_entropy
Computer science problem
various disciplines related to mathematics, including algorithmics, random matrix theory, representation theory, and physics. The longest increasing subsequence
Longest increasing subsequence
Longest_increasing_subsequence
model are random, it is not a random band matrix. The study of random band matrices, however, is related. Bourgade, Paul (2018-06-06), "Random band matrices"
Random_band_matrix
Real square matrix whose columns and rows are orthogonal unit vectors
In linear algebra, an orthogonal matrix or orthonormal matrix Q, is a real-valued square matrix whose columns and rows are orthonormal vectors. One way
Orthogonal_matrix
Age-structured model of population growth
is a generalization of the population growth rate to when a Leslie matrix has random elements which may be correlated. When characterizing the disorder
Leslie_matrix
Italian mathematician
He works in mathematical analysis of many-body quantum systems and random matrix theory. Schlein studied theoretical physics at ETH Zurich and received
Benjamin_Schlein
French physicist (born 1962)
applications of random matrix theory: a short review, Jean-Philippe Bouchaud, Marc Potters, in The Oxford Handbook of Random Matrix Theory Edited by
Jean-Philippe_Bouchaud
British theoretical physicist and mathematician (1923–2020)
PMID 17780673. S2CID 3195432. "A Brownian-Motion Model for the Eigenvalues of a Random Matrix". Journal of Mathematical Physics. 3 (6): 1191–1198. 1962. Bibcode:1962JMP
Freeman_Dyson
Polynomial sequence
systems theory in connection with nonlinear operations on Gaussian noise. random matrix theory in Gaussian ensembles. Hermite polynomials were defined by Pierre-Simon
Hermite_polynomials
Notion in statistics
information is a way of measuring the amount of information that an observable random variable X carries about an unknown parameter θ of a distribution that models
Fisher_information
Software library for similarity search
FAISS provides the following useful facilities: k-means clustering Random-matrix rotations for spreading the variance over all the dimensions without
FAISS
Phenomenon in quantum systems
orbits provide corrections to the universal spectral statistics of the random matrix theory. There are rigorous mathematical theorems on quantum nature of
Quantum_scar
Approximations used in machine learning
Low-rank matrix approximations are essential tools in the application of kernel methods to large-scale learning problems. Kernel methods (for instance
Low-rank matrix approximations
Low-rank_matrix_approximations
Canadian mathematician
Jenssen, and Marcus Michelen on random matrix theory with the paper The singularity probability of a random symmetric matrix is exponentially small. The paper
Julian_Sahasrabudhe
American mathematician
and to noncommutative algebraic geometry, as well as earlier work in random matrix theory, coding theory, lattices, and quantum information theory. Rains
Eric_M._Rains
Concept in probability theory and machine learning
computation, and random matrix theory, a probability distribution over vectors is said to be in isotropic position if its covariance matrix is proportional
Isotropic_position
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
Algorithm
{\displaystyle I} is the identity matrix. The iteration matrix, I − B − 1 Z , {\displaystyle I-B^{-1}Z,} is random, whence the name of this formulation
Kaczmarz_method
Concept in statistical mechanics
mechanics, the Gaussian free field (GFF) is a Gaussian random field, a central model of random surfaces (random height functions). The discrete version can be
Gaussian_free_field
Statistics computed from a sample of data
random vector, sample covariance matrices are positive semi-definite. To prove it, note that for any matrix A {\displaystyle \mathbf {A} } the matrix
Sample_mean_and_covariance
Classification algorithm
transforms a vector of random variables with a known covariance matrix into a set of new variables whose covariance is the identity matrix, meaning that they
Whitening_transformation
Application of physics to the study of economics
with the credit ratings assigned by Moody's. Another good example is random matrix theory, which can be used to identify the noise in financial correlation
Econophysics
The Bohigas–Giannoni–Schmit (BGS) conjecture also known as the random matrix conjecture) for simple quantum mechanical systems (ergodic with a classical
BGS_conjecture
British mathematician
mathematics and mathematical physics, in particular to quantum chaos, random matrix theory and number theory. He read for an MA in physics at New College
Jonathan_Keating
On eigenvalues of random matrices
the study of random matrices, the circular law concerns the distribution of eigenvalues of an n × n {\displaystyle n\times n} random matrix with independent
Circular_law
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
Correlation matrix — a symmetric n×n matrix, formed by the pairwise correlation coefficients of several random variables. Covariance matrix — a symmetric
List_of_named_matrices
Mathematical optimization algorithm
Courtney Paquette, Tom Trogdon, Jeffrey. "Random Matrix Theory and Machine Learning Tutorial". random-matrix-learning.github.io. Retrieved 2023-12-05.{{cite
Conjugate_gradient_method
invariants of spectral curves. It has applications in enumerative geometry, random matrix theory, mathematical physics, string theory, knot theory. The topological
Topological_recursion
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
Statistical distribution of complex random variables
normal complex random vectors that are circularly symmetric are of particular interest because they are fully specified by the covariance matrix Γ {\displaystyle
Complex_normal_distribution
physicist, known in particular for contributions to random matrix theory, the spectral theory of random Schrödinger operators, statistical mechanics, and
Leonid_Pastur
Generalization of beta distribution
p\times p} orthogonal matrix, then H U H T ∼ B ( a , b ) . {\displaystyle HUH^{T}\sim B(a,b).} Also, if H {\displaystyle H} is a random orthogonal p × p {\displaystyle
Matrix variate beta distribution
Matrix_variate_beta_distribution
Measure of the joint variability
random variables with finite second moments, its auto-covariance matrix (also known as the variance–covariance matrix or simply the covariance matrix)
Covariance
Idea that small causes can have large effects
developed by Martin Gutzwiller and John B. Delos and co-workers. The random matrix theory and simulations with quantum computers prove that some versions
Butterfly_effect
American engineer (born 1968)
condensed matter theory. His published work focused on superconductivity, random matrix theory, the quantum hall effect, and particle astrophysics. In 1998
Safi_Bahcall
2021 film by Lana Wachowski
The Matrix Resurrections is a 2021 American science fiction action film co-produced and directed by Lana Wachowski, who co-wrote the screenplay with David
The_Matrix_Resurrections
Statistical measure of how far values spread from their average
independent random variables (if it exists). The formula states that the variance of a sum is equal to the sum of all elements in the covariance matrix of the
Variance
Perturbation theory Probability theory Proof theory Queue theory Ramsey theory Random matrix theory Representation theory Ring theory Scheme theory Semigroup theory
List_of_mathematical_theories
Tree-based ensemble machine learning methods
Random forests or random decision forests is an ensemble learning method for classification, regression and other tasks that works by creating a multitude
Random_forest
Average value of a random variable
E[X]_{i}=E[X_{i}]} . Similarly, one may define the expected value of a random matrix X {\displaystyle X} with components X i j {\displaystyle X_{ij}} by
Expected_value
Stochastic point process in mathematics
and other inference tasks. Such processes arise as important tools in random matrix theory, combinatorics, physics, machine learning, and wireless network
Determinantal_point_process
Determinant of a product of rectangular matrices
\det(AB)=\left|{\begin{matrix}1&1\\3&1\end{matrix}}\right|\cdot \left|{\begin{matrix}1&1\\3&1\end{matrix}}\right|+\left|{\begin{matrix}1&2\\1&-1\end{matrix}}\right|\cdot
Cauchy–Binet_formula
Matrix in which most of the elements are zero
In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. There is no strict
Sparse_matrix
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
Mathematical result
Afonso Bandeira that can be found at the following link. Construct a random matrix A ∼ N ( 0 , 1 ) k × n {\displaystyle A\sim {\mathcal {N}}(0,1)^{k\times
Johnson–Lindenstrauss_lemma
Random Matrix Ensembles (Parallel implementation of circular ensemble generation)". doi:10.5281/zenodo.579642. "Bristol: A Python package for Random Matrix
Circular_ensemble
American mathematician
mathematical physics, quantum integrable systems, stochastic PDEs, and random matrix theory. He is particularly known for work related to the Kardar–Parisi–Zhang
Ivan_Corwin
Concept in statistics
In statistics, an exchangeable sequence of random variables (also sometimes interchangeable) is a sequence X1, X2, X3, ... (which may be finitely or infinitely
Exchangeable_random_variables
Israeli mathematician
present in energy levels of quantum chaotic systems and described by random matrix theory. Together with Peter Sarnak, he has formulated the Quantum Unique
Zeev_Rudnick
Mathematical conjecture about the Riemann zeta function
as the pair correlation distribution for the eigenvalues of a random Hermitian matrix. These distributions are of importance in physics — the eigenstates
Hilbert–Pólya_conjecture
cross-correlation matrix of two random vectors is a matrix containing as elements the cross-correlations of all pairs of elements of the random vectors. The
Cross-correlation_matrix
American mathematician
covering: quantum field theory, statistical mechanics, Brownian motion, random matrix theory, general nonrelativistic quantum mechanics (including N-body
Barry_Simon
Special function in the physical sciences
describes the law of largest eigenvalues in Random matrix. Due to the intimate connection of random matrix theory with the Kardar–Parisi–Zhang equation
Airy_function
Canadian and American theoretical physicist
random-matrix and related methods for studying quantum chaos." He was a co-author of a highly cited (2700 citations) review article on random-matrix physics
James_Bruce_French
as feature extraction method. The RM consists in generation of a random matrix that is multiplied by each original vector and result in a reduced vector
Random_mapping
American mathematician (1980–2020)
Algebra (Cambridge University Press, 2018). She is also the author of The Random Matrix Theory of the Classical Compact Groups (Cambridge University Press,
Elizabeth_Meckes
Type of signal in signal processing
prescribed covariance matrix. Conversely, a random vector with known covariance matrix can be transformed into a white random vector by a suitable whitening
White_noise
Surname list
Madan Lal Mehta (1932–2006), theoretical physicist in the field of random matrix theory Mehli Mehta, (1908–2002): musician; founder of the Bombay Philharmonic
Mehta
Statistical relationship
correlation matrix of n {\displaystyle n} random variables X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} is the n × n {\displaystyle n\times n} matrix C {\displaystyle
Correlation
RANDOM MATRIX
RANDOM MATRIX
Surname or Lastname
English
English : patronymic from Rand 1.
Male
English
Medieval form of English Randolf, RANDAL means "shield-wolf."
Male
Hungarian
 Variant spelling of Hungarian András, ANDOR means "man; warrior." Compare with another form of Andor.
Boy/Male
English
Son of Rand.
Surname or Lastname
English
English : variant spelling of Randall.Americanized spelling of Randel.
Male
Norwegian
 Norwegian form of Old Norse Arnþórr, ANDOR means "eagle of Thor." Compare with another form of Andor.
Female
English
Short form of English Miranda, RANDA means "worthy of admiration."Â
Surname or Lastname
English
English : variant of Rand 1, from the Old French oblique case.
Surname or Lastname
English
English : probably a variant of Crandon, a habitational name from Crandon in Somerset or Crandean in Falmer, Sussex. Compare Grandin.
Surname or Lastname
English
English : variant of Ransom.
Male
Scandinavian
 Scandinavian form of Old Norse Randolfr, RANDOLF means "shield-wolf." Compare with another form of Randolf.
Male
English
Pet form of English Randall and Randolph, both RANDY means "shield-wolf." Compare with feminine Randy.
Male
English
 Variant spelling of Middle English Randulf, RANDOLF means "shield-wolf." Compare with other forms of Randolf.
Surname or Lastname
English
English : variant of Brandon.
Surname or Lastname
English (chiefly East Anglia)
English (chiefly East Anglia) : patronymic from the Middle English personal name Rand(e) (see Rand 1).
Surname or Lastname
English
English : unexplained; perhaps a variant of Francom.
Surname or Lastname
English or Scottish
English or Scottish : unexplained. Possibly, as Black suggests, a reduced form of Langdon.French : from the old Germanic personal name element Lando (see Land), via the oblique case, Landonis.
Female
English
Pet form of English Miranda, RANDY means "worthy of admiration."Â Compare with masculine Randy.Â
Boy/Male
English American
Son of Rand.
Female
English
Variant spelling of English Randy, RANDI means "worthy of admiration."
RANDOM MATRIX
RANDOM MATRIX
Boy/Male
Indian, Jain
Cleft
Boy/Male
Hindu, Indian
New
Boy/Male
American, Australian, French, German, Latin
Patrician; Noble; Form of Patrick
Female
English
Feminine form of English Adrian, ADRIANNE means "from Hadria."
Girl/Female
Indian
Gift, Present
Boy/Male
Indian, Punjabi, Sikh
Exalted Meditation
Girl/Female
English American
and Kayla, meaning: keeper of the keys; pure.
Boy/Male
Arabic, Muslim
Revered; Servant of the Sublime One
Surname or Lastname
English
English : variant of Wool.Americanized form of Jewish Wollman or German Wollmann (see Wollman).
Boy/Male
Hindu, Indian
God Perumal
RANDOM MATRIX
RANDOM MATRIX
RANDOM MATRIX
RANDOM MATRIX
RANDOM MATRIX
n.
Extra hazard; chance; accident; random.
n.
Distance to which a missile is cast; range; reach; as, the random of a rifle ball.
imp. & p. p.
of Ransom
adv.
At random; hit or miss. (Obs.)
a.
Cruising at random on the ocean.
n.
To redeem from captivity, servitude, punishment, or forfeit, by paying a price; to buy out of servitude or penalty; to rescue; to deliver; as, to ransom prisoners from an enemy.
v. i.
To go or stray at random.
adv.
In a random manner.
n.
Anything driven at random.
n.
The release of a captive, or of captured property, by payment of a consideration; redemption; as, prisoners hopeless of ransom.
p. pr. & vb. n.
of Ransom
n.
Random.
a.
Going at random or by chance; done or made at hazard, or without settled direction, aim, or purpose; hazarded without previous calculation; left to chance; haphazard; as, a random guess.
n.
To exact a ransom for, or a payment on.
v. i.
To wander at random; to scatter.
n.
Ransom.
n.
A roving motion; course without definite direction; want of direction, rule, or method; hazard; chance; -- commonly used in the phrase at random, that is, without a settled point of direction; at hazard.
v. i.
To extend or grow at random.
n.
Ransom; release.