Spectral radius

In mathematics, the spectral radius of a square matrix or a bounded linear operator is the largest absolute value of its eigenvalues (i.e. supremum among the absolute values of the elements in its spectrum). It is sometimes denoted by ρ(·).

Matrices

Let λ₁, ..., λ_n be the (real or complex) eigenvalues of a matrix A ∈ C^n×n. Then its spectral radius ρ(A) is defined as:

${\displaystyle \rho (A)=\max \left\{|\lambda _{1}|,\dotsc ,|\lambda _{n}|\right\}.}$

The condition number of ${\displaystyle A}$ can be expressed using the spectral radius as ${\displaystyle \rho (A)\rho (A^{-1})}$.

The spectral radius is a sort of infimum of all norms of a matrix. On the one hand, ${\displaystyle \rho (A)\leqslant \|A\|}$ for every natural matrix norm ${\displaystyle \|\cdot \|}$, and on the other hand, Gelfand's formula states that ${\displaystyle \rho (A)=\lim _{k\to \infty }\|A^{k}\|^{1/k}}$; both these results are shown below. However, the spectral radius does not necessarily satisfy ${\displaystyle \|A\mathbf {v} \|\leqslant \rho (A)\|\mathbf {v} \|}$ for arbitrary vectors ${\displaystyle \mathbf {v} \in \mathbb {C} ^{n}}$. To see why, let ${\displaystyle r>1}$ be arbitrary and consider the matrix ${\displaystyle C_{r}={\begin{pmatrix}0&r^{-1}\\r&0\end{pmatrix}}}$. The characteristic polynomial of ${\displaystyle C_{r}}$ is ${\displaystyle \lambda ^{2}-1}$, hence its eigenvalues are ${\displaystyle \pm 1}$, and thus ${\displaystyle \rho (C_{r})=1}$. However ${\displaystyle C_{r}\mathbf {e} _{1}=r\mathbf {e} _{2}}$, so ${\displaystyle \|C_{r}\mathbf {e} _{1}\|=r>1=\rho (C_{r})\|\mathbf {e} _{1}\|}$ for ${\displaystyle \|\cdot \|}$ being any ${\displaystyle \ell ^{p}}$ norm on ${\displaystyle \mathbb {C} ^{n}}$ . What still allows ${\displaystyle \|C_{r}^{k}\|^{1/k}\to 1}$ as ${\displaystyle k\to \infty }$ is that ${\displaystyle C_{r}^{2}=I}$, making ${\displaystyle \|C_{r}^{k}\|^{1/k}\leqslant \|C_{r}\|^{1/k}=r^{1/k}\to 1}$ as ${\displaystyle k\to \infty }$.

${\displaystyle \|A\mathbf {v} \|\leqslant \rho (A)\|\mathbf {v} \|}$ for all ${\displaystyle \mathbf {v} \in \mathbb {C} ^{n}}$

does hold when ${\displaystyle A}$ is a Hermitian matrix and ${\displaystyle \|\cdot \|}$ is the Euclidean norm.

Graphs

The spectral radius of a finite graph is defined to be the spectral radius of its adjacency matrix.

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:

${\displaystyle \ell ^{2}(G)=\left\{f:V(G)\to \mathbf {R} \$ :\ \sum \nolimits _{v\in V(G)}\left\|f(v)^{2}\right\|<\infty \right\}.}

Let γ be the adjacency operator of G:

${\displaystyle {\begin{cases}\gamma$ :\ell ^{2}(G)\to \ell ^{2}(G)\\(\gamma f)(v)=\sum _{(u,v)\in E(G)}f(u)\end{cases}}}

The spectral radius of G is defined to be the spectral radius of the bounded linear operator γ.

Upper bound

Upper bounds for spectral radius of a matrix

The following proposition shows a simple yet useful upper bound for the spectral radius of a matrix:

Proposition. Let A ∈ C^n×n with spectral radius ρ(A) and a consistent matrix norm ||⋅||. Then for each integer ${\displaystyle k\geqslant 1}$:

${\displaystyle \rho (A)\leq \|A^{k}\|^{\frac {1}{k}}.}$

Proof

Let (v, λ) be an eigenvector-eigenvalue pair for a matrix A. By the sub-multiplicative property of the matrix norm, we get:

${\displaystyle |\lambda |^{k}\|\mathbf {v} \|=\|\lambda ^{k}\mathbf {v} \|=\|A^{k}\mathbf {v} \|\leq \|A^{k}\|\cdot \|\mathbf {v} \|}$

and since v ≠ 0 we have

${\displaystyle |\lambda |^{k}\leq \|A^{k}\|}$

and therefore

${\displaystyle \rho (A)\leq \|A^{k}\|^{\frac {1}{k}}.}$

Upper bounds for spectral radius of a graph

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. For instance, if

${\displaystyle {\frac {(k-2)(k-3)}{2}}\leq m-n\leq {\frac {k(k-3)}{2}}}$

where ${\displaystyle 3\leq k\leq n}$ is an integer, then[1]

${\displaystyle \rho (G)\leq {\sqrt {2m-n-k+{\frac {5}{2}}+{\sqrt {2m-2n+{\frac {9}{4}}}}}}}$

Power sequence

Theorem

The spectral radius is closely related to the behaviour of the convergence of the power sequence of a matrix; namely, the following theorem holds:

Theorem. Let A ∈ C^n×n with spectral radius ρ(A). Then ρ(A) < 1 if and only if

${\displaystyle \lim _{k\to \infty }A^{k}=0.}$

On the other hand, if ρ(A) > 1, ${\displaystyle \lim _{k\to \infty }\|A^{k}\|=\infty }$. The statement holds for any choice of matrix norm on C^n×n.

Proof of theorem

Assume the limit in question is zero, we will show that ρ(A) < 1. Let (v, λ) be an eigenvector-eigenvalue pair for A. Since A^kv = λ^kv we have:

${\displaystyle {\begin{aligned}0&=\left(\lim _{k\to \infty }A^{k}\right)\mathbf {v} \\&=\lim _{k\to \infty }\left(A^{k}\mathbf {v} \right)\\&=\lim _{k\to \infty }\lambda ^{k}\mathbf {v} \\&=\mathbf {v} \lim _{k\to \infty }\lambda ^{k}\end{aligned}}}$

and, since by hypothesis v ≠ 0, we must have

${\displaystyle \lim _{k\to \infty }\lambda ^{k}=0}$

which implies |λ| < 1. Since this must be true for any eigenvalue λ, we can conclude ρ(A) < 1.

Now assume the radius of A is less than 1. From the Jordan normal form theorem, we know that for all A ∈ C^n×n, there exist V, J ∈ C^n×n with V non-singular and J block diagonal such that:

${\displaystyle A=VJV^{-1}}$

with

${\displaystyle J={\begin{bmatrix}J_{m_{1}}(\lambda _{1})&0&0&\cdots &0\\0&J_{m_{2}}(\lambda _{2})&0&\cdots &0\\\vdots &\cdots &\ddots &\cdots &\vdots \\0&\cdots &0&J_{m_{s-1}}(\lambda _{s-1})&0\\0&\cdots &\cdots &0&J_{m_{s}}(\lambda _{s})\end{bmatrix}}}$

where

${\displaystyle J_{m_{i}}(\lambda _{i})={\begin{bmatrix}\lambda _{i}&1&0&\cdots &0\\0&\lambda _{i}&1&\cdots &0\\\vdots &\vdots &\ddots &\ddots &\vdots \\0&0&\cdots &\lambda _{i}&1\\0&0&\cdots &0&\lambda _{i}\end{bmatrix}}\in \mathbf {C} ^{m_{i}\times m_{i}},1\leq i\leq s.}$

It is easy to see that

${\displaystyle A^{k}=VJ^{k}V^{-1}}$

and, since J is block-diagonal,

${\displaystyle J^{k}={\begin{bmatrix}J_{m_{1}}^{k}(\lambda _{1})&0&0&\cdots &0\\0&J_{m_{2}}^{k}(\lambda _{2})&0&\cdots &0\\\vdots &\cdots &\ddots &\cdots &\vdots \\0&\cdots &0&J_{m_{s-1}}^{k}(\lambda _{s-1})&0\\0&\cdots &\cdots &0&J_{m_{s}}^{k}(\lambda _{s})\end{bmatrix}}}$

Now, a standard result on the k-power of an ${\displaystyle m_{i}\times m_{i}}$ Jordan block states that, for ${\displaystyle k\geq m_{i}-1}$:

${\displaystyle J_{m_{i}}^{k}(\lambda _{i})={\begin{bmatrix}\lambda _{i}^{k}&{k \choose 1}\lambda _{i}^{k-1}&{k \choose 2}\lambda _{i}^{k-2}&\cdots &{k \choose m_{i}-1}\lambda _{i}^{k-m_{i}+1}\\0&\lambda _{i}^{k}&{k \choose 1}\lambda _{i}^{k-1}&\cdots &{k \choose m_{i}-2}\lambda _{i}^{k-m_{i}+2}\\\vdots &\vdots &\ddots &\ddots &\vdots \\0&0&\cdots &\lambda _{i}^{k}&{k \choose 1}\lambda _{i}^{k-1}\\0&0&\cdots &0&\lambda _{i}^{k}\end{bmatrix}}}$

Thus, if ${\displaystyle \rho (A)<1}$ then for all i ${\displaystyle |\lambda _{i}|<1}$. Hence for all i we have:

${\displaystyle \lim _{k\to \infty }J_{m_{i}}^{k}=0}$

which implies

${\displaystyle \lim _{k\to \infty }J^{k}=0.}$

Therefore,

${\displaystyle \lim _{k\to \infty }A^{k}=\lim _{k\to \infty }VJ^{k}V^{-1}=V\left(\lim _{k\to \infty }J^{k}\right)V^{-1}=0}$

On the other side, if ${\displaystyle \rho (A)>1}$, there is at least one element in J which doesn't remain bounded as k increases, so proving the second part of the statement.

Gelfand's formula

Theorem

The next theorem gives the spectral radius as a limit of matrix norms.

Theorem (Gelfand's Formula; 1941). For any matrix norm ||⋅||, we have

${\displaystyle \rho (A)=\lim _{k\to \infty }\left\|A^{k}\right\|^{\frac {1}{k}}.}$[2].

Proof

For any ε > 0, first we construct the following two matrices:

${\displaystyle A_{\pm }={\frac {1}{\rho (A)\pm \varepsilon }}A.}$

Then:

${\displaystyle \rho \left(A_{\pm }\right)={\frac {\rho (A)}{\rho (A)\pm \varepsilon }},\qquad \rho (A_{+})<1<\rho (A_{-}).}$

First we apply the previous theorem to A₊:

${\displaystyle \lim _{k\to \infty }A_{+}^{k}=0.}$

That means, by the sequence limit definition, there exists N₊ ∈ N such that for all k ≥ N₊,

${\displaystyle {\begin{aligned}\left\|A_{+}^{k}\right\|<1\end{aligned}}}$

so

${\displaystyle {\begin{aligned}\left\|A^{k}\right\|^{\frac {1}{k}}<\rho (A)+\varepsilon .\end{aligned}}}$

Applying the previous theorem to A₋ implies ${\displaystyle \|A_{-}^{k}\|}$ is not bounded and there exists N₋ ∈ N such that for all k ≥ N₋,

${\displaystyle {\begin{aligned}\left\|A_{-}^{k}\right\|>1\end{aligned}}}$

so

${\displaystyle {\begin{aligned}\left\|A^{k}\right\|^{\frac {1}{k}}>\rho (A)-\varepsilon .\end{aligned}}}$

Let N = max{N₊, N₋}, then we have:

${\displaystyle \forall \varepsilon >0\quad \exists N\in \mathbf {N} \quad \forall k\geq N\quad \rho (A)-\varepsilon <\left\|A^{k}\right\|^{\frac {1}{k}}<\rho (A)+\varepsilon }$

which, by definition, is

${\displaystyle \lim _{k\to \infty }\left\|A^{k}\right\|^{\frac {1}{k}}=\rho (A).}$

Gelfand corollaries

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

${\displaystyle \rho (A_{1}\cdots A_{n})\leq \rho (A_{1})\cdots \rho (A_{n}).}$

Actually, in case the norm is 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:

${\displaystyle \forall \varepsilon >0,\exists N\in \mathbf {N} ,\forall k\geq N\quad \rho (A)\leq \|A^{k}\|^{\frac {1}{k}}<\rho (A)+\varepsilon }$

which, by definition, is

${\displaystyle \lim _{k\to \infty }\left\|A^{k}\right\|^{\frac {1}{k}}=\rho (A)^{+},}$

where the + means that the limit is approached from above.

Example

Consider the matrix

${\displaystyle A={\begin{bmatrix}9&-1&2\\-2&8&4\\1&1&8\end{bmatrix}}}$

whose eigenvalues are 5, 10, 10; by definition, ρ(A) = 10. In the following table, the values of ${\displaystyle \|A^{k}\|^{\frac {1}{k}}}$ for the four most used norms are listed versus several increasing values of k (note that, due to the particular form of this matrix,${\displaystyle \|.\|_{1}=\|.\|_{\infty }}$):

k	${\displaystyle \|.\|_{1}=\|.\|_{\infty }}$	${\displaystyle \|.\|_{F}}$	${\displaystyle \|.\|_{2}}$
1	14	15.362291496	10.681145748
2	12.649110641	12.328294348	10.595665162
3	11.934831919	11.532450664	10.500980846
4	11.501633169	11.151002986	10.418165779
5	11.216043151	10.921242235	10.351918183
${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$
10	10.604944422	10.455910430	10.183690042
11	10.548677680	10.413702213	10.166990229
12	10.501921835	10.378620930	10.153031596
${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$
20	10.298254399	10.225504447	10.091577411
30	10.197860892	10.149776921	10.060958900
40	10.148031640	10.112123681	10.045684426
50	10.118251035	10.089598820	10.036530875
${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$
100	10.058951752	10.044699508	10.018248786
200	10.029432562	10.022324834	10.009120234
300	10.019612095	10.014877690	10.006079232
400	10.014705469	10.011156194	10.004559078
${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$
1000	10.005879594	10.004460985	10.001823382
2000	10.002939365	10.002230244	10.000911649
3000	10.001959481	10.001486774	10.000607757
${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$
10000	10.000587804	10.000446009	10.000182323
20000	10.000293898	10.000223002	10.000091161
30000	10.000195931	10.000148667	10.000060774
${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$	${\displaystyle \vdots }$
100000	10.000058779	10.000044600	10.000018232

Bounded linear operators

For a bounded linear operator A and the operator norm ||·||, again we have

${\displaystyle \rho (A)=\lim _{k\to \infty }\|A^{k}\|^{\frac {1}{k}}.}$

A bounded operator (on a complex Hilbert space) is called a spectraloid operator if its spectral radius coincides with its numerical radius. An example of such an operator is a normal operator.

Notes and references

1. Guo, Ji-Ming; Wang, Zhi-Wen; Li, Xin (2019). "Sharp upper bounds of the spectral radius of a graph". Discrete Mathematics. 342 (9): 2559–2563. doi:10.1016/j.disc.2019.05.017.
2. The formula holds for any Banach algebra; see Lemma IX.1.8 in Dunford and Schwartz 1963 harvnb error: no target: CITEREFDunford_and_Schwartz1963 (help) and Lax 2002, pp. 195–197

• Dunford, Nelson; Schwartz, Jacob (1963), Linear operators II. Spectral Theory: Self Adjoint Operators in Hilbert Space, Interscience Publishers, Inc.
• Lax, Peter D. (2002), Functional Analysis, Wiley-Interscience, ISBN 0-471-55604-1

See also

• Spectral gap
• The Joint spectral radius is a generalization of the spectral radius to sets of matrices.
• Spectrum of a matrix
• Spectral abscissa