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Concept
Spectral radius
Summary
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. The spectral radius is often denoted by ρ(·).
Let λ1, ..., λn be the eigenvalues of a matrix A ∈ Cn×n. The spectral radius of A is defined as
The spectral radius can be thought of as an infimum of all norms of a matrix. Indeed, on the one hand, for every natural matrix norm ; and on the other hand, Gelfand's formula states that . Both of these results are shown below.
However, the spectral radius does not necessarily satisfy for arbitrary vectors . To see why, let be arbitrary and consider the matrix
The characteristic polynomial of is , so its eigenvalues are and thus . However, . As a result,
As an illustration of Gelfand's formula, note that as , since if is even and if is odd.
A special case in which for all is when is a Hermitian matrix and 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,
In the context of a bounded linear operator A on a Banach space, the eigenvalues need to be replaced with the elements of the spectrum of the operator, i.e. the values for which 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:
Gelfand's formula, also known as the spectral radius formula, also holds for bounded linear operators: letting 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.
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.
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