Spectral radius

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:

ρ ( A ) = max { | λ 1 | , ⋯ , | λ n | } .

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 k ∈ N: ρ ( A ) ≤ ∥ A k ∥ 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:

| λ | k ∥ v ∥ = ∥ λ k v ∥ = ∥ A k v ∥ ≤ ∥ A k ∥ ⋅ ∥ v ∥

and since v ≠ 0 we have

| λ | k ≤ ∥ A k ∥

and therefore

ρ ( A ) ≤ ∥ A k ∥ 1 k .

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 lim k → ∞ A k = 0. On the other hand, if ρ(A) > 1, lim k → ∞ ∥ A k ∥ = ∞ .

The statement holds for any choice of matrix norm on C^n×n.

Proof. 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:

0 = ( lim k → ∞ A k ) v = lim k → ∞ ( A k v ) = lim k → ∞ λ k v = v lim k → ∞ λ k

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

lim k → ∞ λ 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:

A = V J V − 1

with

J = [ J m 1 ( λ 1 ) 0 0 ⋯ 0 0 J m 2 ( λ 2 ) 0 ⋯ 0 ⋮ ⋯ ⋱ ⋯ ⋮ 0 ⋯ 0 J m s − 1 ( λ s − 1 ) 0 0 ⋯ ⋯ 0 J m s ( λ s ) ]

where

J m i ( λ i ) = [ λ i 1 0 ⋯ 0 0 λ i 1 ⋯ 0 ⋮ ⋮ ⋱ ⋱ ⋮ 0 0 ⋯ λ i 1 0 0 ⋯ 0 λ i ] ∈ C m i × m i , 1 ≤ i ≤ s .

It is easy to see that

A k = V J k V − 1

and, since J is block-diagonal,

J k = [ J m 1 k ( λ 1 ) 0 0 ⋯ 0 0 J m 2 k ( λ 2 ) 0 ⋯ 0 ⋮ ⋯ ⋱ ⋯ ⋮ 0 ⋯ 0 J m s − 1 k ( λ s − 1 ) 0 0 ⋯ ⋯ 0 J m s k ( λ s ) ]

Now, a standard result on the k-power of an m i × m i Jordan block states that, for k ≥ m i − 1 :

J m i k ( λ i ) = [ λ i k ( k 1 ) λ i k − 1 ( k 2 ) λ i k − 2 ⋯ ( k m i − 1 ) λ i k − m i + 1 0 λ i k ( k 1 ) λ i k − 1 ⋯ ( k m i − 2 ) λ i k − m i + 2 ⋮ ⋮ ⋱ ⋱ ⋮ 0 0 ⋯ λ i k ( k 1 ) λ i k − 1 0 0 ⋯ 0 λ i k ]

Thus, if ρ ( A ) < 1 then for all i | λ i | < 1 . Hence for all i we have:

lim k → ∞ J m i k = 0

which implies

lim k → ∞ J k = 0.

Therefore,

lim k → ∞ A k = lim k → ∞ V J k V − 1 = V ( lim k → ∞ J k ) V − 1 = 0

On the other side, if ρ ( 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 (Gelfand's Formula; 1941). For any matrix norm ||⋅||, we have ρ ( A ) = lim k → ∞ ∥ A k ∥ 1 k .

Proof

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

A ± = 1 ρ ( A ) ± ε A .

Then:

ρ ( A ± ) = ρ ( A ) ρ ( A ) ± ε , ρ ( A + ) < 1 < ρ ( A − ) .

First we apply the previous theorem to A₊:

lim k → ∞ A + k = 0.

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

∥ A + k ∥ < 1

so

∥ A k ∥ 1 k < ρ ( A ) + ε .

Applying the previous theorem to A₋ implies ∥ A − k ∥ is not bounded and there exists N₋ ∈ N such that for all k ≥ N+,

∥ A − k ∥ > 1

so

∥ A k ∥ 1 k > ρ ( A ) − ε .

Let N = max{N₊, N₋}, then we have:

∀ ε > 0 ∃ N ∈ N ∀ k ≥ N ρ ( A ) − ε < ∥ A k ∥ 1 k < ρ ( A ) + ε

which, by definition, is

lim k → ∞ ∥ A k ∥ 1 k = ρ ( A ) .

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

ρ ( A 1 ⋯ A n ) ≤ ρ ( A 1 ) ⋯ ρ ( 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:

∀ ε > 0 , ∃ N ∈ N , ∀ k ≥ N ρ ( A ) ≤ ∥ A k ∥ 1 k < ρ ( A ) + ε

which, by definition, is

lim k → ∞ ∥ A k ∥ 1 k = ρ ( A ) + ,

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

Example

Consider the matrix

A = [ 9 − 1 2 − 2 8 4 1 1 8 ]

whose eigenvalues are 5, 10, 10; by definition, ρ(A) = 10. In the following table, the values of ∥ A k ∥ 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, ∥ . ∥ 1 = ∥ . ∥ ∞ ):

Bounded Linear Operators

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

ρ ( A ) = lim k → ∞ ∥ A k ∥ 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.

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:

ℓ 2 ( G ) = { f : V ( G ) → R   :   ∑ v ∈ V ( G ) ∥ f ( v ) 2 ∥ < ∞ } .

Let γ be the adjacency operator of G:

{ γ : ℓ 2 ( G ) → ℓ 2 ( G ) ( γ f ) ( v ) = ∑ ( u , v ) ∈ E ( G ) f ( u )

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