Logarithmic norm

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.
Original definition.
Let formula_1 be a square matrix and formula_2 be an induced matrix norm. The associated logarithmic norm formula_3 of formula_1 is defined
Here formula_6 is the identity matrix of the same dimension as formula_1, and formula_8 is a real, positive number. The limit as formula_9 equals formula_10, and is in general different from the logarithmic norm formula_11, as formula_12 for all matrices.
The matrix norm formula_13 is always positive if formula_14, but the logarithmic norm formula_11 may also take negative values, e.g. when formula_1 is negative definite. Therefore, the logarithmic norm does not satisfy the axioms of a norm. 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 formula_18 is formula_11. This is expressed by the differential inequality 
where formula_21 is the upper right Dini derivative. 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 formula_23 associated to the differential equation formula_24 is bounded by
for all formula_26.
Alternative definitions.
If the vector norm is an inner product norm, as in a Hilbert space, then the logarithmic norm is the smallest number formula_11 such that for all formula_28 
Unlike the original definition, the latter expression also allows formula_1 to be unbounded. Thus differential operators too can have logarithmic norms, allowing the use of the logarithmic norm both in algebra and in analysis. The modern, extended theory therefore prefers a definition based on inner products or duality. Both the operator norm and the logarithmic norm are then associated with extremal values of quadratic forms as follows: 
Properties.
Basic properties of the logarithmic norm of a matrix include:
Example logarithmic norms.
The logarithmic norm of a matrix can be calculated as follows for the three most common norms. In these formulas, formula_44 represents the element on the formula_45th row and formula_46th column of a matrix formula_1.
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 formula_52. This also means that every eigenvalue formula_53 of formula_1 satisfies
More generally, the logarithmic norm is related to the numerical range of a matrix.
A matrix with formula_56 is positive definite, and one with formula_57 is negative definite. Such matrices have inverses. 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. Some results only hold for inner product norms, however. For example, if formula_59 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.
Applications in stability theory and numerical analysis.
The logarithmic norm plays an important role in the stability analysis of a continuous dynamical system formula_62. Its role is analogous to that of the matrix norm for a discrete dynamical system formula_63.
In the simplest case, when formula_1 is a scalar complex constant formula_65, the discrete dynamical system has stable solutions when formula_66, while the differential equation has stable solutions when formula_67. When formula_1 is a matrix, the discrete system has stable solutions if formula_69. In the continuous system, the solutions are of the form formula_70. They are stable if formula_71 for all formula_72, which follows from property 7 above, if formula_73. In the latter case, formula_74 is a Lyapunov function for the system.
Runge–Kutta methods for the numerical solution of formula_62 replace the differential equation by a discrete equation formula_76, where the rational function formula_59 is characteristic of the method, and formula_8 is the time step size. If formula_79 whenever formula_80, then a stable differential equation, having formula_73, will always result in a stable (contractive) numerical method, as formula_82. 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.
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 formula_83 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. In the latter, equality is attained for the function formula_86, implying that the constant formula_87 is the best possible. Thus
for the differential operator formula_89, which implies that
As an operator satisfying formula_91 is called elliptic, the logarithmic norm quantifies the (strong) ellipticity of formula_92. Thus, if formula_1 is strongly elliptic, then formula_94, and is invertible given proper data.
If a finite difference method is used to solve formula_95, the problem is replaced by an algebraic equation formula_96. The matrix formula_97 will typically inherit the ellipticity, i.e., formula_98, showing that formula_97 is positive definite and therefore invertible.
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 formula_101 is the least upper bound Lipschitz constant of formula_102, and formula_103 is the greatest lower bound Lipschitz constant; and
where formula_105 and formula_106 are in the domain formula_107 of formula_102. Here formula_109 is the least upper bound logarithmic Lipschitz constant of formula_102, and formula_103 is the greatest lower bound logarithmic Lipschitz constant. It holds that formula_112 (compare above) and, analogously, formula_113, where formula_114 is defined on the image of formula_102.
For nonlinear operators that are Lipschitz continuous, it further holds that
If formula_102 is differentiable and its domain formula_107 is convex, then
Here formula_121 is the Jacobian matrix of formula_102, linking the nonlinear extension to the matrix norm and logarithmic norm.
An operator having either formula_123 or formula_124 is called uniformly monotone. An operator satisfying formula_125 is called contractive. This extension offers many connections to fixed point theory, and critical point theory.
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).