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Wikipedia / Spectral radius

In 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 linear operator is the supremum of the absolute values of the elements of its spectrum.

Definition

Matrices

Let be the eigenvalues of a matrix . The spectral radius of is defined as

for every natural matrix norm. Both of these results are shown below.

be arbitrary and consider the matrix

  1. .

The characteristic polynomial. As a result,

is odd.

is a Hermitian matrix is the Euclidean norm. This is because any Hermitian Matrix is diagonalizable by a unitary matrix, and unitary matrices preserve vector length. As a result,

Bounded linear operators

In the context of a bounded linear operator on a Banach space, the eigenvalues need to be replaced with the elements of the spectrum of the operator is not bijective. We denote the spectrum by

The spectral radius is then defined as the supremum of the magnitudes of the elements of the spectrum:

denote 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. 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 such that the degree of every vertex of the graph is smaller than ). In this case, for the graph define:

Let be the adjacency operator of :

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

Upper bounds

Upper bounds on the spectral radius of a matrix

The following proposition gives simple yet useful upper bounds on the spectral radius of a matrix.

Proposition. Let with spectral radius and a sub-multiplicative matrix norm :

Proof

Let be an eigenvector-eigenvalue pair for a matrix A. By the sub-multiplicativity of the matrix norm, we get:

Since , we have

and therefore

concluding the proof.

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

is an integer, then

Symmetric matrices

denotes the spectral norm is symmetric, this inequality is tight:

Theorem.

Proof

be the eigenpairs of A. Due to the symmetry of A are orthonormal which implies that

it follows that

and

Power sequence

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.

Theorem. Let with spectral radius . Then if and only if

On the other hand, if . The statement holds for any choice of matrix norm on .

Proof

goes to infinity. We will show that . Let be an eigenvector-eigenvalue pair for A. Since , we have

Since by hypothesis, we must have

, we can conclude that .

Now, assume the radius of is less than . From the Jordan normal form theorem, we know that for all , there exist with non-singular and block diagonal such that:

with

where

It is easy to see that

and, since is block-diagonal,

Now, a standard result on the :

then for all . Hence for all we have:

which implies

Therefore,

, there is at least one element in that does not remain bounded as increases, thereby proving the second part of the statement.

Gelfand's formula

Gelfand's formula, named after Israel Gelfand, gives the spectral radius as a limit of matrix norms.

Theorem

For any matrix norm we have

  1. .

Moreover, in the case of a consistent).

Proof

For any , let us define the two following matrices:

Thus,

We start by applying the previous theorem on limits of power sequences to :

This shows the existence of such that, for all ,

Therefore,

is not bounded and that there exists such that, for all ,

Therefore,

Let }. Then,

that is,

This concludes the proof.

Corollary

are matrices that all commute, then

Numerical example

Consider the matrix

whose eigenvalues are ; by definition, ):

Notes and references

Bibliography

See also