Spectral radius
In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues.[1] More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectrum.Let λ1, ..., λn be the eigenvalues of a matrix A ∈ Cn×n.; and on the other hand, Gelfand's formula states thatbe arbitrary and consider the matrix The characteristic polynomial ofAs a result, As an illustration of Gelfand's formula, note thatThis is because any Hermitian Matrix is diagonalizable by a unitary matrix, and unitary matrices preserve vector length.As a result, In the context of a bounded linear operator A on a Banach space, the eigenvalues need to be replaced with the elements of the spectrum of the operator, i.e. the valuesWe denote the spectrum by The spectral radius is then defined as the supremum of the magnitudes of the elements of the spectrum: Gelfand's formula, also known as the spectral radius formula, also holds for bounded linear operators: lettingdenote the operator norm, we have A bounded operator (on a complex Hilbert space) is called a spectraloid operator if its spectral radius coincides with its numerical radius.This definition extends to the case of infinite graphs with bounded degrees of vertices (i.e. there exists some real number C such that the degree of every vertex of the graph is smaller than C).In this case, for the graph G define: Let γ be the adjacency operator of G: The spectral radius of G is defined to be the spectral radius of the bounded linear operator γ.The following proposition gives simple yet useful upper bounds on the spectral radius of a matrix.Let A ∈ Cn×n with spectral radius ρ(A) and a consistent matrix norm ||⋅||.: Proof Let (v, λ) be an eigenvector-eigenvalue pair for a matrix A.By the sub-multiplicativity of the matrix norm, we get: Since v ≠ 0, we have and therefore concluding the proof.There are many upper bounds for the spectral radius of a graph in terms of its number n of vertices and its number m of edges.is symmetric, this inequality is tight: Theorem.The spectral radius is closely related to the behavior of the convergence of the power sequence of a matrix; namely as shown by the following theorem.The statement holds for any choice of matrix norm on Cn×n.Let (v, λ) be an eigenvector-eigenvalue pair for A.Since Akv = λkv, we have Since v ≠ 0 by hypothesis, we must have which impliesFrom the Jordan normal form theorem, we know that for all A ∈ Cn×n, there exist V, J ∈ Cn×n with V non-singular and J block diagonal such that: with where It is easy to see that and, since J is block-diagonal, Now, a standard result on the k-power of an, there is at least one element in J that does not remain bounded as k increases, thereby proving the second part of the statement.Gelfand's formula, named after Israel Gelfand, gives the spectral radius as a limit of matrix norms.For any ε > 0, let us define the two following matrices: Thus, We start by applying the previous theorem on limits of power sequences to A+: This shows the existence of N+ ∈ N such that, for all k ≥ N+, Therefore, Similarly, the theorem on power sequences implies thatis not bounded and that there exists N− ∈ N such that, for all k ≥ N−, Therefore, Let N = max{N+, N−}.Gelfand's formula yields a bound on the spectral radius of a product of commuting matrices: ifare matrices that all commute, then Consider the matrix whose eigenvalues are 5, 10, 10; by definition, ρ(A) = 10.for the four most used norms are listed versus several increasing values of k (note that, due to the particular form of this matrix,
mathematicssquare matrixeigenvaluesbounded linear operatorsupremumspectrumcharacteristic polynomialHermitian matrixEuclidean normdiagonalizableunitary matrixBanach spacespectrum of the operatoroperator normnumerical radiusnormal operatoradjacency matrixeigenvectoreigenvaluesymmetricorthonormalJordan normal formIsrael Gelfandmatrix normconsistentBanach algebraLax, Peter D.Spectral gapJoint spectral radiusSpectrum of a matrixSpectral abscissaFunctional analysistopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledCompleteLocally convexReflexiveSeparableHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointBoundedCompactHilbert–SchmidtNormalTrace classTransposeUnboundedUnitaryC*-algebraSpectrum of a C*-algebraOperator algebraGroup algebra of a locally compact groupVon Neumann algebraInvariant subspace problemMahler's conjectureHardy spaceSpectral theory of ordinary differential equationsHeat kernelIndex theoremCalculus of variationsFunctional calculusIntegral linear operatorJones polynomialTopological quantum field theoryNoncommutative geometryRiemann hypothesisDistributionGeneralized functionsApproximation propertyBalanced setChoquet theoryWeak topologyBanach–Mazur distanceTomita–Takesaki theorySpectral theory*-algebrasInvolution/*-algebraB*-algebraNoncommutative topologyProjection-valued measureOperator spaceGelfand–Mazur theoremGelfand–Naimark theoremGelfand representationPolar decompositionSingular value decompositionSpectral theoremSpectral theory of normal C*-algebrasIsospectraloperatorHermitian/Self-adjointKrein–Rutman theoremNormal eigenvalueSpectral asymmetryDecomposition of a spectrumContinuousDirect integralDiscreteBorel functional calculusMin-max theoremPositive operator-valued measureRiesz projectorRigged Hilbert spaceSpectral theory of compact operatorsAmenable Banach algebraApproximate identityBanach function algebraDisk algebraNuclear C*-algebra