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| In [[mathematics]], the '''spectral radius''' of a [[matrix (mathematics)|square matrix]] or a [[bounded linear operator]] is the [[supremum]] among the [[absolute value]]s of the elements in its [[spectrum of a matrix|spectrum]], which is sometimes denoted by ρ(·).
| | Hello! My name is Charley. <br>It is a little about myself: I live in Italy, my city of Vallo Di Caluso. <br>It's called often Northern or cultural capital of TO. I've married 1 years ago.<br>I have two children - a son (Roxanna) and the daughter (Mac). We all like Fossil hunting.<br><br>Here is my homepage [http://budi.kepegawaian.unej.ac.id/?author=726 Hostgator Discount] |
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| ==Matrices==
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| Let λ<sub>1</sub>, ..., λ<sub>''n''</sub> be the ([[real number|real]] or [[complex number|complex]]) eigenvalues of a matrix ''A'' ∈ '''C'''<sup>''n'' × ''n''</sup>. Then its spectral radius ρ(''A'') is defined as:
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| :<math>\rho(A) \overset{\underset{\mathrm{def}}{}}{=} \max_i(|\lambda_i|)</math>
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| The following lemma shows a simple yet useful upper bound for the spectral radius of a matrix:
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| '''Lemma''': Let <math>A \in \mathbb{C}^{n \times n}</math> be a complex-valued matrix, ρ(''A'') its spectral radius and ||·|| a [[matrix norm#Consistent_norms|consistent matrix norm]]; then, for each ''k'' ∈ '''N''':
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| <math>\rho(A)\leq \|A^k\|^{1/k}.</math>
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| ''Proof'': Let ('''v''', λ) be an [[eigenvector]]-[[eigenvalue]] pair for a matrix ''A''. By the sub-multiplicative property of the matrix norm, we get:
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| :<math>|\lambda|^k\|\mathbf{v}\| = \|\lambda^k \mathbf{v}\| = \|A^k \mathbf{v}\| \leq \|A^k\|\cdot\|\mathbf{v}\|</math>
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| and since '''v''' ≠ 0 for each λ we have
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| :<math>|\lambda|^k\leq \|A^k\|</math>
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| and therefore
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| :<math>\rho(A)\leq \|A^k\|^{1/k}\,\,\square</math>
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| The spectral radius is closely related to the behaviour of the convergence of the power sequence of a matrix; namely, the following theorem holds:
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| '''Theorem''': Let ''A'' ∈ '''C'''<sup>''n'' × ''n''</sup> be a complex-valued matrix and ρ(''A'') its spectral radius; then
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| :<math>\lim_{k \to \infty}A^k=0</math> if and only if <math>\rho(A)<1.</math>
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| Moreover, if ρ(''A'')>1, <math>\|A^k\|</math> is not bounded for increasing k values.
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| ''Proof'':
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| <math>\left(\lim_{k \to \infty}A^k = 0 \Rightarrow \rho(A) < 1\right)</math>
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| Let ('''v''', λ) be an [[eigenvector]]-[[eigenvalue]] pair for matrix ''A''. Since
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| :<math>A^k\mathbf{v} = \lambda^k\mathbf{v},</math>
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| we have:
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| :<math>\begin{align}
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| 0 &= \left(\lim_{k \to \infty}A^k\right)\mathbf{v} \\
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| &= \lim_{k \to \infty}A^k\mathbf{v} \\
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| &= \lim_{k \to \infty}\lambda^k\mathbf{v} \\
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| &= \mathbf{v}\lim_{k \to \infty}\lambda^k
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| \end{align}</math>
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| and, since by hypothesis '''v''' ≠ 0, we must have
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| :<math>\lim_{k \to \infty}\lambda^k = 0</math>
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| which implies |λ| < 1. Since this must be true for any eigenvalue λ, we can conclude ρ(''A'') < 1.
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| <math>\left(\rho(A)<1 \Rightarrow \lim_{k \to \infty}A^k = 0\right)</math>
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| From the [[Jordan normal form]] theorem, we know that for any complex valued matrix <math>A \in \mathbb{C}^{n \times n}</math>, a non-singular matrix <math>V \in \mathbb{C}^{n \times n}</math> and a block-diagonal matrix <math>J \in \mathbb{C}^{n \times n}</math> exist such that:
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| :<math>A = VJV^{-1}</math>
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| with
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| :<math>J=\begin{bmatrix}
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| J_{m_1}(\lambda_1) & 0 & 0 & \cdots & 0 \\
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| 0 & J_{m_2}(\lambda_2) & 0 & \cdots & 0 \\
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| \vdots & \cdots & \ddots & \cdots & \vdots \\
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| 0 & \cdots & 0 & J_{m_{s-1}}(\lambda_{s-1}) & 0 \\
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| 0 & \cdots & \cdots & 0 & J_{m_s}(\lambda_s)
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| \end{bmatrix}</math>
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| where
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| :<math>J_{m_i}(\lambda_i)=\begin{bmatrix}
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| \lambda_i & 1 & 0 & \cdots & 0 \\
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| 0 & \lambda_i & 1 & \cdots & 0 \\
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| \vdots & \vdots & \ddots & \ddots & \vdots \\
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| 0 & 0 & \cdots & \lambda_i & 1 \\
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| 0 & 0 & \cdots & 0 & \lambda_i
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| \end{bmatrix}\in \mathbb{C}^{m_i,m_i}, 1\leq i\leq s.</math>
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| It is easy to see that
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| :<math>A^k=VJ^kV^{-1}</math>
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| and, since <math>J</math> is block-diagonal,
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| :<math>J^k=\begin{bmatrix}
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| J_{m_1}^k(\lambda_1) & 0 & 0 & \cdots & 0 \\
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| 0 & J_{m_2}^k(\lambda_2) & 0 & \cdots & 0 \\
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| \vdots & \cdots & \ddots & \cdots & \vdots \\
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| 0 & \cdots & 0 & J_{m_{s-1}}^k(\lambda_{s-1}) & 0 \\
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| 0 & \cdots & \cdots & 0 & J_{m_s}^k(\lambda_s)
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| \end{bmatrix}</math>
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| Now, a standard result on the <math>k</math>-power of an <math>m_i \times m_i</math> Jordan block states that, for <math>k \geq m_i-1</math>:
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| :<math>J_{m_i}^k(\lambda_i)=\begin{bmatrix}
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| \lambda_i^k & {k \choose 1}\lambda_i^{k-1} & {k \choose 2}\lambda_i^{k-2} & \cdots & {k \choose m_i-1}\lambda_i^{k-m_i+1} \\
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| 0 & \lambda_i^k & {k \choose 1}\lambda_i^{k-1} & \cdots & {k \choose m_i-2}\lambda_i^{k-m_i+2} \\
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| \vdots & \vdots & \ddots & \ddots & \vdots \\
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| 0 & 0 & \cdots & \lambda_i^k & {k \choose 1}\lambda_i^{k-1} \\
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| 0 & 0 & \cdots & 0 & \lambda_i^k
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| \end{bmatrix}</math>
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| Thus, if <math>\rho(A) < 1</math> then <math>|\lambda_i| < 1 \forall i</math>, so that
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| :<math>\lim_{k \to \infty}J_{m_i}^k=0\ \forall i</math>
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| which implies
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| :<math>\lim_{k \to \infty}J^k = 0.</math>
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| Therefore,
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| :<math>\lim_{k \to \infty}A^k=\lim_{k \to \infty}VJ^kV^{-1}=V(\lim_{k \to \infty}J^k)V^{-1}=0</math>
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| On the other side, if <math>\rho(A)>1</math>, there is at least one element in <math>J</math> which doesn't remain bounded as k increases, so proving the second part of the statement. <math>\square</math>
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| ==Theorem (Gelfand's formula, 1941)==
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| For any [[matrix norm]] ||·||, we have
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| :<math>\rho(A)=\lim_{k \to \infty}\|A^k\|^{1/k}.</math>
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| In other words, Gelfand's formula shows how the spectral radius of ''A'' gives the asymptotic growth rate of the norm of ''A''<sup>''k''</sup>:
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| :<math>\|A^k\|\sim\rho(A)^k</math> for <math>k\rightarrow \infty.\,</math>
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| ''Proof'': For any ε > 0, consider the matrix
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| :<math>\tilde{A}=(\rho(A)+\epsilon)^{-1}A.</math>
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| Then, obviously,
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| :<math>\rho(\tilde{A}) = \frac{\rho(A)}{\rho(A)+\epsilon} < 1</math>
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| and, by the previous theorem,
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| :<math>\lim_{k \to \infty}\tilde{A}^k=0.</math>
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| That means, by the sequence limit definition, a natural number ''N<sub>1</sub>'' ∈ '''N''' exists such that
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| :<math>\forall k\geq N_1 \Rightarrow \|\tilde{A}^k\| < 1</math>
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| which in turn means:
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| :<math>\forall k\geq N_1 \Rightarrow \|A^k\| < (\rho(A)+\epsilon)^k</math>
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| or
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| :<math>\forall k\geq N_1 \Rightarrow \|A^k\|^{1/k} < (\rho(A)+\epsilon).</math>
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| Let's now consider the matrix
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| :<math>\check{A}=(\rho(A)-\epsilon)^{-1}A.</math>
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| Then, obviously,
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| :<math>\rho(\check{A}) = \frac{\rho(A)}{\rho(A)-\epsilon} > 1</math>
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| and so, by the previous theorem,<math>\|\check{A}^k\|</math> is not bounded. | |
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| This means a natural number ''N<sub>2</sub>'' ∈ '''N''' exists such that
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| :<math>\forall k\geq N_2 \Rightarrow \|\check{A}^k\| > 1</math>
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| which in turn means:
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| :<math>\forall k\geq N_2 \Rightarrow \|A^k\| > (\rho(A)-\epsilon)^k</math>
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| or
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| :<math>\forall k\geq N_2 \Rightarrow \|A^k\|^{1/k} > (\rho(A)-\epsilon).</math>
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| Taking
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| :<math>N:=\max(N_1,N_2)</math>
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| and putting it all together, we obtain:
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| :<math>\forall \epsilon>0, \exists N\in\mathbb{N}: \forall k\geq N \Rightarrow \rho(A)-\epsilon < \|A^k\|^{1/k} < \rho(A)+\epsilon</math>
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| which, by definition, is
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| :<math>\lim_{k \to \infty}\|A^k\|^{1/k} = \rho(A).\,\,\square</math>
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| Gelfand's formula leads directly to a bound on the spectral radius of a product of finitely many matrices, namely assuming that they all commute we obtain
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| <math>
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| \rho(A_1 A_2 \ldots A_n) \leq \rho(A_1) \rho(A_2)\ldots \rho(A_n).
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| </math>
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| Actually, in case the norm is [[matrix norm|consistent]], the proof shows more than the thesis; in fact, using the previous lemma, we can replace in the limit definition the left lower bound with the spectral radius itself and write more precisely:
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| ::<math>\forall \epsilon>0, \exists N\in\mathbb{N}: \forall k\geq N \Rightarrow \rho(A) \leq \|A^k\|^{1/k} < \rho(A)+\epsilon</math>
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| :which, by definition, is
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| :<math>\lim_{k \to \infty}\|A^k\|^{1/k} = \rho(A)^+.</math>
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| '''Example''': Let's consider the matrix
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| :<math>A=\begin{bmatrix}
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| 9 & -1 & 2\\
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| -2 & 8 & 4\\
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| 1 & 1 & 8
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| \end{bmatrix}</math>
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| whose eigenvalues are 5, 10, 10; by definition, its spectral radius is ρ(''A'')=10. In the following table, the values of <math>\|A^k\|^{1/k}</math> for the four most used norms are listed versus several increasing values of k (note that, due to the particular form of this matrix,<math>\|.\|_1=\|.\|_\infty</math>):
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| {| class=wikitable
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| ! ''k''
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| ! <math>\|.\|_1=\|.\|_\infty</math>
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| ! <math>\|.\|_F</math>
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| ! <math>\|.\|_2</math>
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| |-
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| | 1
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| | 14
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| | 15.362291496
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| | 10.681145748
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| |-
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| | 2
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| | 12.649110641
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| | 12.328294348
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| | 10.595665162
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| |-
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| | 3
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| | 11.934831919
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| | 11.532450664
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| | 10.500980846
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| |-
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| | 4
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| | 11.501633169
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| | 11.151002986
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| | 10.418165779
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| |-
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| | 5
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| | 11.216043151
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| | 10.921242235
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| | 10.351918183
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| |-
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| |-
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| | 10
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| | 10.604944422
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| | 10.455910430
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| | 10.183690042
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| |-
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| | 11
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| | 10.548677680
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| | 10.413702213
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| | 10.166990229
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| |-
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| | 12
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| | 10.501921835
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| | 10.378620930
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| | 10.153031596
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| |-
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| |-
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| | 20
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| | 10.298254399
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| | 10.225504447
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| | 10.091577411
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| |-
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| | 30
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| | 10.197860892
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| | 10.149776921
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| | 10.060958900
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| |-
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| | 40
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| | 10.148031640
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| | 10.112123681
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| | 10.045684426
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| |-
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| | 50
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| | 10.118251035
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| | 10.089598820
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| | 10.036530875
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| |-
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| |-
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| | 100
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| | 10.058951752
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| | 10.044699508
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| | 10.018248786
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| |-
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| | 200
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| | 10.029432562
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| | 10.022324834
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| | 10.009120234
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| |-
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| | 300
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| | 10.019612095
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| | 10.014877690
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| | 10.006079232
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| |-
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| | 400
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| | 10.014705469
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| | 10.011156194
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| | 10.004559078
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| |-
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| |-
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| | 1000
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| | 10.005879594
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| | 10.004460985
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| | 10.001823382
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| |-
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| | 2000
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| | 10.002939365
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| | 10.002230244
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| | 10.000911649
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| |-
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| | 3000
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| | 10.001959481
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| | 10.001486774
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| | 10.000607757
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| |-
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| |-
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| | 10000
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| | 10.000587804
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| | 10.000446009
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| | 10.000182323
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| |-
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| | 20000
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| | 10.000293898
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| | 10.000223002
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| | 10.000091161
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| |-
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| | 30000
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| | 10.000195931
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| | 10.000148667
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| | 10.000060774
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| |-
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| | <math>\vdots</math>
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| |-
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| | 100000
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| | 10.000058779
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| | 10.000044600
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| | 10.000018232
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| |}
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| ==Bounded linear operators==
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| For a [[bounded linear operator]] ''A'' and the [[operator norm]] ||·||, again we have
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| :<math>\rho(A) = \lim_{k \to \infty}\|A^k\|^{1/k}.</math> | |
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| A bounded operator (on a complex Hilbert space) called a '''spectraloid operator''' if its spectral radius coincides with its [[numerical radius]]. An example of such an operator is a [[normal operator]].
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| ==Graphs==
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| The spectral radius of a finite [[graph (mathematics)|graph]] is defined to be the spectral radius of its [[adjacency matrix]].
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| 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 <math>G</math> let <math> l^2(G) </math> denote the space of functions <math> f \colon V(G) \to {\mathbb R} </math> with <math> \sum_{v \in V(G)} \|f(v)^2\| < \infty </math>. Let <math> \gamma \colon l^2(G) \to l^2(G)</math> be the adjacency operator of <math> G </math>, i.e., <math> (\gamma f)(v) = \sum_{(u,v) \in E(G)} f(u) </math>. The spectral radius of G is defined to be the spectral radius of the bounded linear operator <math>\gamma</math>.
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| ==See also==
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| * [[Spectral gap]]
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| * The [[Joint spectral radius]] is a generalization of the spectral radius to sets of matrices.
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| {{Functional Analysis}}
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| [[Category:Spectral theory]]
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| [[Category:Articles containing proofs]]
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