To obtain an acceptable lower bound, we calculate the spectral radius at each frequency.

There is also the notion of the spectral radius, commonly taken as the largest eigenvalue.

This sum is useful for estimating the spectral radius of .

A basic reproduction number can be calculated as the spectral radius of an appropriate functional operator.

If the spectral radius is less than 1, the system is instead asymptotically stable.

The Joint spectral radius, introduced by Strang in the early 60s.

Furthermore, for square matrices we have the spectral radius formula:

From the previous statement we can derive an estimate of the spectral radius of .

Related to the computability of the joint spectral radius is the following conjecture:

They showed that the joint spectral radius can be used to describe smoothness properties of certain wavelet functions.

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

The Perron number of a graph is the spectral radius of its adjacency matrix.

Their advantage is that they are easy to implement, and in practice, they provide in general the best bounds on the joint spectral radius.

In general, the spectral radius of A is bounded above by the operator norm of A:

For a finite (or more generally compact) set of matrices the joint spectral radius is defined as follows:

In spite of the negative theoretical results on the joint spectral radius computability, methods have been proposed that perform well in practice.

Moreover, the spectrum of an element x is non-empty and satisfies the spectral radius formula:

The operator norm of a normal operator equals its numerical radius and spectral radius.

The joint spectral radius was introduced for its interpretation as a stability condition for discrete-time switching dynamical systems.

A discrete linear time-invariant system is marginally stable if and only if the transfer function's spectral radius is 1.

Now the Sobolev exponent is roughly the order of minus logarithm of the spectral radius of .

The standard convergence condition (for any iterative method) is when the spectral radius of the iteration matrix is less than 1:

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

The peripheral spectrum of an operator is defined as the set of points in its spectrum which have modulus equal to its spectral radius.

A closely related operator is a spectraloid operator: an operator whose spectral radius coincides with its numerical radius.