Spectral radius

Encyclopedia

In mathematics

, the

or a bounded linear operator is the supremum

among the absolute value

s of the elements in its spectrum

, which is sometimes denoted by ρ(·).

or complex

) eigenvalues of a matrix A ∈

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

'Proof: Let (

and since

and therefore

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

if and only if

Moreover, if ρ(A)>1, is not bounded for increasing k values.

Proof:

Let (

we have:

and, since by hypothesis

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

From the Jordan normal form

theorem, we know that for any complex valued matrix , a non-singular matrix and a block-diagonal matrix exist such that:

with

where

It is easy to see that

and, since is block-diagonal,

Now, a standard result on the -power of an Jordan block states that, for :

Thus, if then , so that

which implies

Therefore,

On the other side, if , there is at least one element in which doesn't remain bounded as k increases, so proving the second part of the statement.

In other words, the Gelfand's formula shows how the spectral radius of A gives the asymptotic growth rate of the norm of A

for

Proof: For any ε > 0, consider the matrix

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

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:

whose eigenvalues are 5, 10, 10; by definition, its spectral radius is ρ(A)=10. In the following table, the values of for the four most used norms are listed versus several increasing values of k (note that, due to the particular form of this matrix,):

||·||, again we have

A bounded operator (on a complex Hilbert space) called a

.

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 let denote the space of functions with . Let be the adjacency operator of , i.e., . The spectral radius of G is defined to be the spectral radius of the bounded linear operator .

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the

**spectral radius**of a square matrixMatrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

or a bounded linear operator is the supremum

Supremum

In mathematics, given a subset S of a totally or partially ordered set T, the supremum of S, if it exists, is the least element of T that is greater than or equal to every element of S. Consequently, the supremum is also referred to as the least upper bound . If the supremum exists, it is unique...

among the absolute value

Absolute value

In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...

s of the elements in its spectrum

Spectrum of a matrix

In mathematics, the spectrum of a matrix is the set of its eigenvalues. This notion can be extended to the spectrum of an operator in the infinite-dimensional case.The determinant equals the product of the eigenvalues...

, which is sometimes denoted by ρ(·).

## Matrices

Let λ_{1}, ..., λ_{n}be the (realReal number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

or complex

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

) eigenvalues of a matrix A ∈

**C**^{n × n}. Then its spectral radius ρ(A) is defined as:The following lemma shows a simple yet useful upper bound for the spectral radius of a matrix:

**Lemma**: Let A ∈**C**^{n × n}be a complex-valued matrix, ρ(A) its spectral radius and ||·|| a consistent matrix norm; then, for each k ∈**N**:'Proof: Let (

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

**v**≠ 0 for each λ we haveand therefore

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}be a complex-valued matrix and ρ(A) its spectral radius; thenif and only if

Moreover, if ρ(A)>1, is not bounded for increasing k values.

Proof:

Let (

**v**, λ) be an eigenvector-eigenvalue pair for matrix A. Sincewe have:

and, since by hypothesis

**v**≠ 0, we must havewhich implies |λ| < 1. Since this must be true for any eigenvalue λ, we can conclude ρ(A) < 1.

From the Jordan normal form

Jordan normal form

In linear algebra, a Jordan normal form of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called Jordan matrix, representing the operator on some basis...

theorem, we know that for any complex valued matrix , a non-singular matrix and a block-diagonal matrix exist such that:

with

where

It is easy to see that

and, since is block-diagonal,

Now, a standard result on the -power of an Jordan block states that, for :

Thus, if then , so that

which implies

Therefore,

On the other side, if , there is at least one element in which doesn't remain bounded as k increases, so proving the second part of the statement.

## Theorem (Gelfand's formula, 1941)

For any matrix norm ||·||, we haveIn other words, the Gelfand's formula shows how the spectral radius of A gives the asymptotic growth rate of the norm of A

^{k}:for

Proof: For any ε > 0, consider the matrix

- Then, obviously,

- and, by the previous theorem,

- That means, by the sequence limit definition, a natural number N
_{1}∈**N**exists such that

- which in turn means:

- Let's now consider the matrix

- Then, obviously,

- and so, by the previous theorem, is not bounded.

- This means a natural number N
_{2}∈**N**exists such that

- which in turn means:

- Taking

- and putting it all together, we obtain:

- which, by definition, is

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

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:

- which, by definition, is

**Example**: Let's consider the matrixwhose eigenvalues are 5, 10, 10; by definition, its spectral radius is ρ(A)=10. In the following table, the values of for the four most used norms are listed versus several increasing values of k (note that, due to the particular form of this matrix,):

k | |||
---|---|---|---|

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 |

10 | 10.604944422 | 10.455910430 | 10.183690042 |

11 | 10.548677680 | 10.413702213 | 10.166990229 |

12 | 10.501921835 | 10.378620930 | 10.153031596 |

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 |

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 |

1000 | 10.005879594 | 10.004460985 | 10.001823382 |

2000 | 10.002939365 | 10.002230244 | 10.000911649 |

3000 | 10.001959481 | 10.001486774 | 10.000607757 |

10000 | 10.000587804 | 10.000446009 | 10.000182323 |

20000 | 10.000293898 | 10.000223002 | 10.000091161 |

30000 | 10.000195931 | 10.000148667 | 10.000060774 |

100000 | 10.000058779 | 10.000044600 | 10.000018232 |

## Bounded linear operators

For a bounded linear operator A and the operator normOperator norm

In mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.- Introduction and definition :...

||·||, again we have

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 operatorNormal operator

In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operatorN:H\to Hthat commutes with its hermitian adjoint N*: N\,N^*=N^*N....

.

## Graphs

The spectral radius of a finite graphGraph (mathematics)

In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...

is defined to be the spectral radius of its adjacency matrix

Adjacency matrix

In mathematics and computer science, an adjacency matrix is a means of representing which vertices of a graph are adjacent to which other vertices...

.

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 let denote the space of functions with . Let be the adjacency operator of , i.e., . The spectral radius of G is defined to be the spectral radius of the bounded linear operator .

## See also

- Spectral gapSpectral gapIn mathematics, the spectral gap is the difference between the moduli of the two largest eigenvalues of a matrix or operator; alternately, it is sometimes taken as the smallest non-zero eigenvalue. Various theorems relate this difference to other properties of the system.See:* Expander graph *...
- The Joint spectral radiusJoint spectral radiusIn mathematics, the joint spectral radius is a generalization of the classical notion of spectral radius of a matrix, to sets of matrices. In recent years this notion has found applications in a large number of engineering fields and is still a topic of active research.-General description:The...

is a generalization of the spectral radius to sets of matrices.