Spectral radius

The spectral radius is a concept in linear algebra and functional analysis . The name is explained by the fact that the spectrum of an operator is contained in a circular disk, the radius of which is the spectral radius.

Spectral radius of matrices

definition

The spectral radius of a matrix is the absolute value of the largest eigenvalue of , that is, is defined by ${\displaystyle \rho }$ ${\displaystyle (n\times n)}$ ${\displaystyle A\in \mathbb {C} ^{n\times n}}$ ${\displaystyle A}$ ${\displaystyle \rho }$

${\displaystyle \rho (A):=\max \limits _{1\leq i\leq n}|\lambda _{i}(A)|}$.

The eigenvalues ​​of, at most, which are different, run through . The spectral radius is also noted with instead of with . ${\displaystyle \lambda _{i}}$ ${\displaystyle n}$ ${\displaystyle A}$ ${\displaystyle \operatorname {spr} (A)}$ ${\displaystyle \rho (A)}$

properties

Every induced matrix norm of is at least as large as the spectral radius. If there is an eigenvalue for an eigenvector of , then: ${\displaystyle A}$ ${\displaystyle \lambda }$ ${\displaystyle v}$ ${\displaystyle A}$

${\displaystyle \|A\|=\sup _{x\neq 0}{\frac {\|Ax\|}{\|x\|}}\geq {\frac {\|Av\|}{\|v\|}}={\frac {\|\lambda v\|}{\|v\|}}=|\lambda |{\frac {\|v\|}{\|v\|}}=|\lambda |}$

More generally, this estimate applies to all matrix norms that are compatible with a vector norm. Furthermore, there is at least one induced norm for each (which can be different for different matrices ) so that ${\displaystyle \epsilon >0}$ ${\displaystyle A}$

${\displaystyle \rho (A)\leq \|A\|<\rho (A)+\epsilon }$

applies. Furthermore, for every induced matrix norm:

${\displaystyle \rho (A)=\inf _{n\in \mathbb {N} }{\sqrt[{n}]{\|A^{n}\|}}=\lim _{n\to \infty }{\sqrt[{n}]{\|A^{n}\|}}}$

Applications

The spectral radius is important in splitting processes , for example . If for an invertible matrix , then the iteration converges ${\displaystyle \rho \left(I-B^{-1}A\right)<1}$ ${\displaystyle B\in \mathbb {C} ^{n\times n}}$

${\displaystyle x_{k+1}=B^{-1}\left(B-A\right)x_{k}+B^{-1}b}$

for each starting vector against the exact solution of the linear system of equations . ${\displaystyle x_{0}}$ ${\displaystyle x^{\ast }}$ ${\displaystyle Ax=b}$

Spectral radius in functional analysis

definition

The concept of the spectral radius can also be defined more generally for bounded linear operators on Banach spaces . For a bounded linear operator one defines ${\displaystyle T}$

${\displaystyle \rho (T):=\sup\{|\lambda |:\lambda \in \sigma (T)\}}$,

where denotes the spectrum of . ${\displaystyle \sigma (T)=\{t\in \mathbb {C} \mid T-t\cdot I~{\text{nicht invertierbar}}\}}$ ${\displaystyle T}$

properties

Since the spectrum is closed, the supremum is accepted, so there is a maximum .

You can also show here that

${\displaystyle \rho (T)=\lim _{n\to \infty }{\sqrt[{n}]{\|T^{n}\|}}=\inf _{n\in \mathbb {N} }{\sqrt[{n}]{\|T^{n}\|}}}$

applies, whereby the operator norm means here. ${\displaystyle \|\cdot \|}$

In particular, the spectral radius of an operator is never larger than the norm of the operator, as in the finite dimensional, that is:

${\displaystyle \rho (T)\leq \|T\|}$

If a normal operator is on a Hilbert space , then equality always applies, as the following section will explain in more detail. ${\displaystyle T}$

${\displaystyle C^{\ast }}$-Algebras

If we restrict ourselves more precisely to Hilbert spaces , we can devote ourselves to -algebras . (And thanks to the GNS construction, all algebras can be represented as operator algebras using Hilbert spaces.) In these algebras there is a closer relationship between the spectral radius and the norm for special classes of elements (operators). Be an algebra. Denote by the set of all characters, ie algebraic homomorphisms. This forms a locally compact Hausdorff space and we can do the mapping ${\displaystyle C^{\ast }}$ ${\displaystyle C^{\ast }}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle C^{\ast }}$ ${\displaystyle \Omega ({\mathcal {A}})}$

${\displaystyle a\in {\mathcal {A}}\mapsto {\hat {a}}\in C_{0}(\Omega ({\mathcal {A}}))}$

consider, being through ${\displaystyle {\hat {a}}}$

${\displaystyle {\hat {a}}(\tau ):=\tau (a)}$

is defined. The Gelfand representation theorem for algebra says that this is an isometry as long as it is abelic. For normal (ie commutate) we can consider and obtain the sub- algebra generated by , which is necessarily commutative ${\displaystyle C^{\ast }}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle a\in {\mathcal {A}}}$ ${\displaystyle \{a,a^{\ast }\}}$ ${\displaystyle \{a,a^{\ast }\}}$ ${\displaystyle C^{\ast }}$

${\displaystyle \|a\|=\|{\hat {a}}\|_{\infty }=\sup _{\tau }|\tau (a)|=\sup\{|\lambda |\mid \lambda \in \sigma (a)\}=\rho (a).}$

(A few details still need to be clarified here, e.g. that the spectrum of does not change if one restricts to the sub- algebra . These details are correct and can be found in elementary introductions to -algebra.) ${\displaystyle a}$ ${\displaystyle C^{\ast }}$

Even if not all elements are normal, there is a close relationship between the norm and the spectrum for all elements. Generally applies to everyone ${\displaystyle a\in {\mathcal {A}}}$

${\displaystyle \|a\|={\sqrt {\|a^{\ast }a\|}}={\sqrt {\rho (a^{\ast }a)}},}$

because is self adjoint and therefore normal. ${\displaystyle a^{\ast }a}$

literature

• Stoer: Numerical Mathematics. Springer-Verlag, Berlin 2005, ISBN 3-540-21395-3 .
• Dirk Werner : Functional Analysis. Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6 .