The logarithmic norm was independently introduced by Germund Dahlquist[1] and Sergei Lozinskiĭ in 1958, for square matrices.
[2] The logarithmic norm has a wide range of applications, in particular in matrix theory, differential equations and numerical analysis.
The name logarithmic norm, which does not appear in the original reference, seems to originate from estimating the logarithm of the norm of solutions to the differential equation The maximal growth rate of
Using logarithmic differentiation the differential inequality can also be written showing its direct relation to Grönwall's lemma.
In fact, it can be shown that the norm of the state transition matrix
Unlike the original definition, the latter expression also allows
The modern, extended theory therefore prefers a definition based on inner products or duality.
[5] 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
satisfies More generally, the logarithmic norm is related to the numerical range of a matrix.
The inverse of a negative definite matrix is bounded by Both the bounds on the inverse and on the eigenvalues hold irrespective of the choice of vector (matrix) norm.
is a rational function with the property 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.
The logarithmic norm plays an important role in the stability analysis of a continuous dynamical system
Its role is analogous to that of the matrix norm for a discrete dynamical system
, the discrete dynamical system has stable solutions when
, while the differential equation has stable solutions when
is a matrix, the discrete system has stable solutions if
, will always result in a stable (contractive) numerical method, as
Runge-Kutta methods having this property are called A-stable.
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.
In connection with differential operators it is common to use inner products and integration by parts.
with inner product Then it holds that where the equality on the left represents integration by parts, and the inequality to the right is a Sobolev inequality[citation needed].
is the least upper bound Lipschitz constant of
is the greatest lower bound Lipschitz constant; and where
is the least upper bound logarithmic Lipschitz constant of
is the greatest lower bound logarithmic Lipschitz constant.
For nonlinear operators that are Lipschitz continuous, it further holds that If
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).