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 whose radius is the spectral radius.

• 2.1 Definition
• 2.2 Features

Spectral radius of matrices

Definition

The spectral radius of a matrix is the sum of the largest magnitude eigenvalue of, ie is defined by

It passes through the maximum eigenvalues ​​of. The spectral radius is also listed with instead of.

Properties

Each induced matrix norm of is at least as large as the spectral radius. Indeed, if an eigenvalue to an eigenvector of, then applies

More generally, this estimate is valid for all with a vector norm compatible matrix norms. Further, there is to each at least one induced standard (which may be different for different matrices ), so that

Applies. Moreover, for any induced matrix norm

Applications

The spectral radius is, for example, in splitting procedure is important. If, then the iteration converges

For each starting vector to the exact solution of the linear system of equations.

Spectral radius in the Functional Analysis

Definition

The concept of spectral radius can be defined for general bounded linear operators on Banach spaces. For a bounded linear operator defined one

The spectrum of is.

Properties

It can be shown that the supremum is assumed, that there is a maximum. In addition, one can also show here that

Applies, in which case the operator norm says. The spectral radius of an operator is also, as in the finite, always less than or equal to the norm of the operator, ie. Is a normal operator on a Hilbert space, then always does equality.