In mathematics, the logarithmic norm is a real-valued functional on operators, and is derived from either an inner product, a vector norm, or its induced operator norm. The logarithmic norm was independently introduced by Germund Dahlquist and Sergei Lozinskiĭ in 1958, for square matrices. It has since been extended to nonlinear operators and unbounded operators as well. The logarithmic norm has a wide range of applications, in particular in matrix theory, differential equations and numerical analysis. In the finite dimensional setting it is also referred to as the matrix measure or the Lozinskiĭ measure.
Contents
Original definition
Let
Here
The matrix norm
The maximal growth rate of
where
showing its direct relation to Grönwall's lemma. In fact, it can be shown that the norm of the state transition matrix
for all
Alternative definitions
If the vector norm is an inner product norm, as in a Hilbert space, then the logarithmic norm is the smallest number
Unlike the original definition, the latter expression also allows
Properties
Basic properties of the logarithmic norm of a matrix include:
-
μ ( z I ) = ℜ ( z ) -
μ ( A ) ≤ ∥ A ∥ -
μ ( γ A ) = γ μ ( A ) for scalar γ > 0 -
μ ( A + z I ) = μ ( A ) + ℜ ( z ) -
μ ( A + B ) ≤ μ ( A ) + μ ( B ) -
α ( A ) ≤ μ ( A ) where α ( A ) is the maximal real part of the eigenvalues ofA -
∥ e t A ∥ ≤ e t μ ( A ) for t ≥ 0 -
μ ( A ) < 0 ⇒ ∥ A − 1 ∥ ≤ − 1 / μ ( A )
Example logarithmic norms
The logarithmic norm of a matrix can be calculated as follows for the three most common norms. In these formulas,
Applications in matrix theory and spectral theory
The logarithmic norm is related to the extreme values of the Rayleigh quotient. It holds that
and both extreme values are taken for some vectors
More generally, the logarithmic norm is related to the numerical range of a matrix.
A matrix with
Both the bounds on the inverse and on the eigenvalues hold irrespective of the choice of vector (matrix) norm. Some results only hold for inner product norms, however. For example, if
then, for inner product norms,
Thus the matrix norm and logarithmic norms may be viewed as generalizing the modulus and real part, respectively, from complex numbers to matrices.
Applications in stability theory and numerical analysis
The logarithmic norm plays an important role in the stability analysis of a continuous dynamical system
In the simplest case, when
Runge-Kutta methods for the numerical solution of
Retaining the same form, the results can, under additional assumptions, be extended to nonlinear systems as well as to semigroup theory, where the crucial advantage of the logarithmic norm is that it discriminates between forward and reverse time evolution and can establish whether the problem is well posed. Similar results also apply in the stability analysis in control theory, where there is a need to discriminate between positive and negative feedback.
Applications to elliptic differential operators
In connection with differential operators it is common to use inner products and integration by parts. In the simplest case we consider functions satisfying
Then it holds that
where the equality on the left represents integration by parts, and the inequality to the right is a Sobolev inequality. In the latter, equality is attained for the function
for the differential operator
As an operator satisfying
If a finite difference method is used to solve
These results carry over to the Poisson equation as well as to other numerical methods such as the Finite element method.
Extensions to nonlinear maps
For nonlinear operators the operator norm and logarithmic norm are defined in terms of the inequalities
where
where
For nonlinear operators that are Lipschitz continuous, it further holds that
If
Here
An operator having either
The theory becomes analogous to that of the logarithmic norm for matrices, but is more complicated as the domains of the operators need to be given close attention, as in the case with unbounded operators. Property 8 of the logarithmic norm above carries over, independently of the choice of vector norm, and it holds that
which quantifies the Uniform Monotonicity Theorem due to Browder & Minty (1963).