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Largest absolute value of an operator's eigenvalues
mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded
Spectral_radius
In mathematics, the joint spectral radius is a generalization of the classical notion of spectral radius of a matrix, to sets of matrices. In recent years
Joint_spectral_radius
Norm on a vector space of matrices
vectors), the induced matrix norm is the spectral norm. The spectral norm should not be confused with the spectral radius. The two values do not coincide in
Matrix_norm
Mathematical concept
theory) Cheeger constant (Riemannian geometry) Eigengap Spectral gap (physics) Spectral radius Spectral gap conjecture "Impossible-Seeming Surfaces Confirmed
Spectral_gap
Seventeenth letter of the Greek alphabet
magnetic field. The correlation coefficient of a population parameter The spectral radius of a matrix A {\displaystyle A} denoted as ρ ( A ) {\displaystyle \rho
Rho
Mathematical function often applied to matrices
are the (upper) spectral radius and (upper) spectral abscissa, respectively. Both limits exist, although neither the spectral radius nor the determinant
Logarithmic_norm
Machine learning model training problem
‖ k {\displaystyle \left\|W_{\text{rec}}\right\|^{k}} . So if the spectral radius of W rec {\displaystyle W_{\text{rec}}} is γ < 1 {\displaystyle \gamma
Vanishing_gradient_problem
Tree graph with exactly one vertex of degree >2
Laplacian has always only one eigenvalue equal or greater than 4. The spectral radius of a starlike tree (the largest eigenvalue of its adjacency matrix)
Starlike_tree
Theorem in linear algebra
in absolute value is strictly smaller than r , |λ| < r. Thus, the spectral radius ρ ( A ) {\displaystyle \rho (A)} is equal to r. If the matrix coefficients
Perron–Frobenius_theorem
Type of graph in graph theory
has spectral radius 2 or at most 2. The graphs with spectral radius 2 form two infinite families and three sporadic examples; if we ask for spectral radius
Smith_graph
condition α ( A ) < 0 {\displaystyle \alpha (A)<0} . Spectral radius Deutsch, Emeric (1975). "The Spectral Abscissa of Partitioned Matrices" (PDF). Journal
Spectral_abscissa
Set of a matrix's eigenvalues
eigenvectors. The spectral radius of a square matrix is the largest absolute value of its eigenvalues. In spectral theory, the spectral radius of a bounded
Spectrum_of_a_matrix
Type of algebraic number
closely related case, the Perron number of a graph is defined to be the spectral radius of its adjacency matrix. Any Pisot number or Salem number is a Perron
Perron_number
Collection of mathematical theories
(functional analysis) Spectral radius, Spectrum of an operator, Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Spectral_theory
Measure of the "size" of linear operators
conjugate transpose of the matrix A {\displaystyle A} ). In general, the spectral radius of A {\displaystyle A} is bounded above by the operator norm of A {\displaystyle
Operator_norm
Generative adversarial network variant
matrix, that is, the largest singular value of the matrix, that is, the spectral radius of the matrix (these concepts are the same for matrices, but different
Wasserstein_GAN
Iterative method used to solve a linear system of equations
standard convergence condition (for any iterative method) is when the spectral radius of the iteration matrix is less than 1: ρ ( D − 1 ( L + U ) ) < 1.
Jacobi_method
Concepts from linear algebra
maximum absolute value of all of its eigenvalues. This is known as the spectral radius of the considered matrix. Let λi be an eigenvalue of an n-by-n matrix
Eigenvalues_and_eigenvectors
Computational problems no algorithm can solve
matrices A 1 , … , A m {\displaystyle A_{1},\dots ,A_{m}} , their joint spectral radius is defined as lim sup N → ∞ max i 1 , … , i N ∈ 1 : m ‖ A i 1 ⋯ , A
List_of_undecidable_problems
Particular kind of algebraic structure
(x)} of an element x {\displaystyle x} is non-empty and satisfies the spectral radius formula: sup { | λ | : λ ∈ σ ( x ) } = lim n → ∞ ‖ x n ‖ 1 / n . {\displaystyle
Banach_algebra
Construct for Hermitian matrices
numerical radius is equal to the spectral norm. Still in functional analysis, λ max {\displaystyle \lambda _{\max }} is known as the spectral radius. In the
Rayleigh_quotient
English mathematician
monotone grid class of permutations is equal to the square of the spectral radius of a related bipartite graph. He has also determined bounds on the
David_Bevan_(mathematician)
American mathematician (born 1934)
MA: Wellesley-Cambridge Press. pp. xii+758. MR 0870634. The Joint spectral radius, introduced by Strang and Rota in the early 60s. Strang splitting Roselle
Gilbert_Strang
Set of eigenvalues of a matrix
( ρ ( T ) {\displaystyle \rho (T)} is sometimes used to denote the spectral radius of T {\displaystyle T} ) If λ {\displaystyle \lambda } is an eigenvalue
Spectrum (functional analysis)
Spectrum_(functional_analysis)
In wavelet theory
_{b}^{n}=\mathrm {tr} (T_{h}^{n})} . This sum is useful for estimating the spectral radius of T h {\displaystyle T_{h}} . There is an alternative possibility
Transfer_matrix
Numerical approximation algorithm
differentiable, a sufficient condition for convergence is that the spectral radius of the derivative is strictly bounded by one in a neighborhood of the
Iterative_method
Type of mathematical model used for infectious diseases
DFS(a)=(n^{*}(a),0,0).} A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator. One way to calculate R 0 {\displaystyle
Compartmental models (epidemiology)
Compartmental_models_(epidemiology)
Topics referred to by the same term
coordinates ρ, magnetic coordinates in toroidal and poloidal coordinates ρ, spectral radius of a square matrix Pollard's rho algorithm, for integer factorization
Rho_(disambiguation)
Algorithm for statistical inference on graphical models
convergence condition was formulated by Johnson et al. in 2006, when the spectral radius of the matrix ρ ( I − | D − 1 / 2 A D − 1 / 2 | ) < 1 {\displaystyle
Belief_propagation
Topics referred to by the same term
Perth, Australia Jai Shri Ram, a Hindu slogan and greeting Joint spectral radius, in mathematics JSR Corporation [ja; es], a Japanese chemical business
JSR
Branch of mathematics
{\displaystyle 0\leq p\leq n} , let d p {\displaystyle d_{p}} be the spectral radius of f acting by pullback on the Hodge cohomology group H p , p ( X )
Complex_dynamics
Describing a family of graphs by excluding certain (sub)graphs
Alexandr (2020-03-01). "Forbidden Subgraphs for Graphs of Bounded Spectral Radius, with Applications to Equiangular Lines". Israel Journal of Mathematics
Forbidden graph characterization
Forbidden_graph_characterization
Square matrix used to represent a graph or network
_{1}-\lambda _{2}} is called the spectral gap and it is related to the expansion of G. It is also useful to introduce the spectral radius of A {\displaystyle A}
Adjacency_matrix
Method of solving a linear system of equations
Spectral radius ρ ( C ω ) {\displaystyle \rho (C_{\omega })} of the iteration matrix for the SOR method C ω {\displaystyle C_{\omega }} . The plot shows
Successive_over-relaxation
Theorem of matrix ranks
_{k=0}^{\infty }\left({A}^{-1}{B}\right)^{k}{A}^{-1}} if the spectral radius of A − 1 B {\displaystyle A^{-1}B} is less than one. That is, if the
Woodbury_matrix_identity
Dynamical system which is neither asymptotically stable nor unstable
1 are all distinct. That is, the transfer function's spectral radius is 1. If the spectral radius is less than 1, the system is instead asymptotically
Marginal_stability
Recurrent neural network architecture
lim n → ∞ W n = 0 {\displaystyle \lim _{n\to \infty }W^{n}=0} if the spectral radius of W {\displaystyle W} is smaller than 1. However, with LSTM units
Long_short-term_memory
Element of *-algebra where x* equals x
( a ) = ‖ a ‖ {\displaystyle r(a)=\left\|a\right\|} holds for the spectral radius, because a {\displaystyle a} is normal. According to the continuous
Self-adjoint_element
Mathematical study of linear operators
C*-identity is a very strong requirement. For instance, together with the spectral radius formula, it implies that the C*-norm is uniquely determined by the
Operator_theory
Iterative method used to solve a linear system of equations
N ) {\displaystyle r=\rho (\mathbf {M} ^{-1}\mathbf {N} )} be the spectral radius of M − 1 N {\displaystyle \mathbf {M} ^{-1}\mathbf {N} } . Then the
Gauss–Seidel_method
Matrix equal to its conjugate-transpose
equal to the spectral norm. Still in functional analysis, | λ | max {\displaystyle \left|\lambda \right|_{\max }} is known as the spectral radius. In the context
Hermitian_matrix
Eigenvalue algorithm
the Google PageRank. The method can also be used to calculate the spectral radius (the eigenvalue with the largest magnitude, for a square matrix) by
Power_iteration
Square (0,1) matrix
representing other special number theoretic sums. If we denote the spectral radius of A n {\displaystyle A_{n}} by ρ n {\displaystyle \rho _{n}} , i.e
Redheffer_matrix
Bounded linear operator
therefore, by the spectral theory of compact operators, its spectrum σ(V) = {0}. V is a quasinilpotent operator (that is, the spectral radius, ρ(V), is zero)
Volterra_operator
number (if it exists) of vertices in a graph whose adjacency matrix has spectral radius exactly ( 1 − α ) / ( 2 α ) {\displaystyle (1-\alpha )/(2\alpha )}
Equiangular_lines
Numerical linear algebra algorithm
absolute values of the eigenvalues of S {\displaystyle S} . 2-norm and spectral radius The 2-norm of a matrix A is the norm based on the Euclidean vectornorm;
Jacobi_eigenvalue_algorithm
Dynamical system that exhibits continuous and discrete dynamic behavior
control Variable structure system Variable structure control Joint spectral radius Cyber-physical system Behavior trees (artificial intelligence, robotics
Hybrid_system
V − 1 {\displaystyle FV^{-1}} with the largest absolute value (the spectral radius of F V − 1 {\displaystyle FV^{-1}} ). Next generation matrices can
Next-generation_matrix
Iterative method used to solve a linear system of equations
gives the simplest Chebyshev iteration. This optimal choice yields a spectral radius of min ω ∈ ( 0 , ω max ) ρ ( I − ω A ) = ρ ( I − ω opt A ) = 1 − 2
Modified_Richardson_iteration
Unit of measurement
A solar radius is a unit of distance, commonly understood as 695,700 km and expressed as R ⊙ {\displaystyle R_{\odot }} , used mostly to express the size
Solar_radius
Finite volume method in partial differential equations
F\left(u\left(t\right)\right)}{\partial u}}\right)\ } represents the spectral radius of ∂ F ( u ( t ) ) ∂ u . {\displaystyle {\frac {\partial
MUSCL_scheme
Machine learning technique
generative adversarial networks (GANs) such as the Wasserstein GAN. The spectral radius can be efficiently computed by the following algorithm: INPUT matrix
Normalization (machine learning)
Normalization_(machine_learning)
One of two types of graph
MR 1668059. Lingsheng Shi; Zhipeng Song (2007). "Upper bounds on the spectral radius of book-free and/or K2,l-free graphs". Linear Algebra and Its Applications
Book_(graph_theory)
Real function with secant line between points above the graph itself
function, by the triangle inequality and positive homogeneity. The spectral radius of a nonnegative matrix is a convex function of its diagonal elements
Convex_function
Representation of a matrix as a sum
where ρ ( D ) {\displaystyle \rho (\mathbf {D} )} represents the spectral radius of D, and thus D is a convergent matrix. As a consequence, the iterative
Matrix_splitting
Mathematical representation in functional analysis
numbers f(x) where f ranges over Gelfand space of A. Together with the spectral radius formula, this shows that  is a subset of the unit ball of A* and as
Gelfand_representation
Italian-American mathematician (1932–1999)
Rota's conjecture Rota's basis conjecture Rota–Baxter algebra Joint spectral radius, introduced by Rota in the early 1960s Cyclotomic identity Necklace
Gian-Carlo_Rota
Classification of stars based on spectral properties
stellar classification is the classification of stars based on their spectral characteristics. Electromagnetic radiation from the star is analyzed by
Stellar_classification
(on a complex Hilbert space) continuous linear operator
operator norm of a normal operator equals its numerical radius[clarification needed] and spectral radius. A normal operator coincides with its Aluthge transform
Normal_operator
Analytic space in mathematics
{\displaystyle A} gives a point in the spectrum of A {\displaystyle A} . The spectral radius of f , {\displaystyle f,} ρ ( f ) = lim n → ∞ ‖ f n ‖ 1 n {\displaystyle
Berkovich_space
Generalization of the exponential function
\|T(t_{0})\|<1} , There exists a t 1 > 0 {\textstyle t_{1}>0} such that the spectral radius of T ( t 1 ) {\textstyle T(t_{1})} is strictly smaller than 1, There
C0-semigroup
physically stable systems (, ), the method can give stable solutions if the spectral radius . A method is called unconditionally stable if the condition is satisfied
Composite methods for structural dynamics
Composite_methods_for_structural_dynamics
Topological complex vector space
C*-identity is a very strong requirement. For instance, together with the spectral radius formula, it implies that the C*-norm is uniquely determined by the
C*-algebra
American mathematician (1931–2019)
generated by a jump distribution with support G. He showed that the spectral radius equals the exponential decay rate of the return probabilities. He showed
Harry_Kesten
Function that maps matrices to matrices
of the matrix function beyond the set of matrices with spectral radius smaller than the radius of convergence of the power series. Note that there is
Analytic_function_of_a_matrix
Generalization of the Perron–Frobenius theorem to Banach spaces
meaning that T ( K ) ⊂ K {\displaystyle T(K)\subset K} , and that its spectral radius r ( T ) {\displaystyle r(T)} is strictly positive. Then r ( T ) {\displaystyle
Krein–Rutman_theorem
Matrix that converges to zero matrix
convergent matrix. Note that ρ(T) = 1/4, where ρ(T) represents the spectral radius of T, since 1/4 is the only eigenvalue of T. Let T be an n × n matrix
Convergent_matrix
Matrix in linear algebra
particular, the entries on the main diagonal must be nonnegative. the spectral radius ρ(A) is an eigenvalue of A. Every copositive matrix of order less than
Copositive_matrix
Important atomic emission spectra
been divided into a number of spectral series, with wavelengths given by the Rydberg formula. These observed spectral lines are due to the electron making
Hydrogen_spectral_series
holds. Every normal element is a normaloid element, i.e. the spectral radius r ( a ) {\displaystyle r(a)} equals the norm of a {\displaystyle a}
Normal_element
Mathematical concept
Banach algebra to a uniform algebra on X. This result follows from the spectral radius formula and the Gelfand representation. (Gamelin 2005, p. 25) Gamelin
Uniform_algebra
Irish mathematician (1948–2006)
joint papers between Murphy and West, Finbarr Holland picks out a spectral radius formula for special mention in an obituary of Gerard that appeared
Gerard_Murphy_(mathematician)
Property of a dynamical system where solutions near an equilibrium point remain so
asymptotically stable (in fact, exponentially stable) if the joint spectral radius of the set { A 1 , … , A m } {\displaystyle \{A_{1},\dots ,A_{m}\}}
Lyapunov_stability
Block diagonal matrix of Jordan blocks
f (A) converges absolutely for every square matrix whose spectral radius is less than the radius of convergence of f around 0 and is uniformly convergent
Jordan_matrix
Measure of centrality in a network based on nodal influence
nonnegative attenuation factor which must be smaller than the inverse of the spectral radius of A {\displaystyle A} . The original definition by Katz used a constant
Katz_centrality
Soviet mathematician (1913–2009)
co-authored with Sergei Fomin; Gelfand's formula, which expresses the spectral radius as a limit of matrix norms. the Gelfand representation in Banach algebra
Israel_Gelfand
operator is a spectraloid operator: an operator whose spectral radius coincides with its numerical radius. In fact, an operator T is convexoid if and only
Convexoid_operator
Computer-aided geometric design
PIA is related to the properties of the collocation matrix. If the spectral radius of the iteration matrix I − B {\displaystyle \mathbf {I} -\mathbf {B}
Progressive-iterative approximation method
Progressive-iterative_approximation_method
Measure of radiant energy over surface area
dimensions and units: spectral irradiance of a frequency spectrum is measured in watts per square metre per hertz (W⋅m−2⋅Hz−1), while spectral irradiance of a
Irradiance
Large self-illuminated object in space
of specific frequencies. In 1865, Secchi began classifying stars into spectral types. The modern version of the stellar classification scheme was developed
Star
of normality of a matrix. In particular, for normal matrices A with spectral radius less than 1, one has that 𝒦(A) = 1. Similarly, for normal matrices
Kreiss_matrix_theorem
Matrix used to describe the transitions of a Markov chain
{\boldsymbol {\pi }}P={\boldsymbol {\pi }}.} It can be shown that the spectral radius of any stochastic matrix is one. By the Gershgorin circle theorem,
Stochastic_matrix
Homotopic map of a graph
unique Perron–Frobenius eigenvalue λ(f) ≥ 1 which is equal to the spectral radius of M(f). One then defines a number of different moves on topological
Train_track_map
Dim, low mass stars on the main sequence
the Sun, with masses about 7.5% that of the Sun. These red dwarfs have spectral types of L0 to L2. There is some overlap with the properties of brown dwarfs
Red_dwarf
Clustering methods
In multivariate statistics, spectral clustering techniques make use of the spectrum (eigenvalues) of the similarity matrix of the data to perform dimensionality
Spectral_clustering
Relation of a matrix of variables between two points in time
real or complex) have an absolute value which is less than 1 (i.e., a spectral radius less than 1). Assume that the equation has been put in the homogeneous
Matrix_difference_equation
space Sobolev space Tsirelson space ba space Uniform norm Matrix norm Spectral radius Normed division algebra Stone–Weierstrass theorem Banach algebra *-algebra
List of functional analysis topics
List_of_functional_analysis_topics
Concept in differential equation mathematics
1} , here ρ ( A ( Δ t ) ) {\displaystyle \rho (A(\Delta t))} is the spectral radius of the update matrix A ( Δ t ) {\displaystyle A(\Delta t)} . For the
Newmark-beta_method
Branch of functional analysis
Sometimes spectral projections inherit properties from their parent operators. For example if T is a positive matrix with spectral radius r then the
Holomorphic functional calculus
Holomorphic_functional_calculus
Super-Earth orbiting LHS 1140
small red dwarf, LHS 1140. It is 18.4% the mass and 21.6% the radius of the Sun with a spectral type of M4.5V. The temperature of LHS 1140 is 3,096 K (2,823 °C;
LHS_1140_b
dimensions, a JB algebra is isomorphic to a Euclidean Jordan algebra. The spectral radius on a JB algebra defines an equivalent norm also satisfying the axioms
Jordan_operator_algebra
Theory of stochastic partial differential equations
dynamical systems theory, a system can be characterized as chaotic if the spectral radius of the finite-time GTO is larger than unity. Under this condition,
Supersymmetric theory of stochastic dynamics
Supersymmetric_theory_of_stochastic_dynamics
Quasar and Lyman-alpha blob in the constellations of Canes Venatici and Coma Berenices
both NV and CIV emission lines in order to calculate the widths of the Hβ spectral line of at least 29 quasars, including TON 618, as a direct measurement
TON_618
Subatomic particle with positive charge
energy of the gluon fields that bind the quarks together. The proton charge radius is around 0.841 fm but two different kinds of measurements give slightly
Proton
Let A ∈ G L ( n , C ) {\displaystyle A\in GL(n,\mathbb {C} )} with spectral radius ρ ( A ) < 1 {\displaystyle \rho (A)<1} . Set F ( z ) = A z {\displaystyle
Fatou–Bieberbach_domain
Spherical collection of stars
means of standard radii: the core radius (rc), the half-light radius (rh), and the tidal or Jacobi radius (rt). The radius can be expressed as a physical
Globular_cluster
Measure of the size of atomic nuclei
The charge radius of an atomic nucleus tells its size. The nucleus (center) of an atom is incredibly tiny. A nucleus, as with an atom, is actually a hazy
Charge_radius
\mathbb {R} ^{n\times n}} . Let ρ ( X ) {\displaystyle \rho (X)} be the spectral radius of a matrix X {\displaystyle X} . Let T J = D − 1 ( L + U ) {\displaystyle
Stein-Rosenberg_theorem
at epoch Radius comparison: RE, R🜨 - Radius compared to Earth RJ, R♃ - Radius compared to Jupiter RS, R☉ - Radius compared to The Sun Spectral comparison:
List of common astronomy symbols
List_of_common_astronomy_symbols
Binary star in the constellation Dorado
largest known star with a well-defined radius, calculated to be around 1,540 times that of the Sun (R☉). At this radius, an object travelling at the speed
WOH_G64
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