In mathematics, the logarithmic norm is a real-valued functional on operators, constructed from either a vector norm or an inner product, or directly from the induced operator norm. It quantifies key notions such as positive/negative definiteness in matrix theory, uniformly coercive or monotone vector fields in nonlinear analysis, and strong ellipticity in differential operators on function spaces, subject to specific boundary conditions. The logarithmic norm has a wide range of applications in matrix theory, stability theory for initial and boundary value problems in differential equations, in nonlinear analysis, applied mathematics, and numerical analysis. Comprehensive treatments of the theory and its applications can be found in recent monographs, on which this article is based.
History, original definition, and classical notation The logarithmic norm was introduced in 1958, independently by Germund Dahlquist and Sergei Lozinskiĭ for square matrices and bounded linear operators. The original purpose was to estimate solutions to linear differential equations x ˙ = A x + r {\displaystyle {\dot {x}}=Ax+r} , to construct sufficient conditions for stability, and to obtain norm bounds of perturbations due to the forcing function r {\displaystyle r} . Let A {\displaystyle A} be a square matrix and ‖ ⋅ ‖ {\displaystyle \|\cdot \|} be the operator norm induced by a given vector norm. The associated logarithmic norm μ [ A ] {\displaystyle \mu [A]} is defined by
μ [ A ] = lim h → 0 + ‖ I + h A ‖ − 1 h , {\displaystyle \mu [A]=\lim \limits _{h\rightarrow 0^{+}}{\frac {\|I+hA\|-1}{h}}\,,}
where h {\displaystyle h} is real and I {\displaystyle I} is the identity matrix of the same dimension as A {\displaystyle A} . The limit exists on account of the convexity of the matrix norm. The term logarithmic norm is due to Lozinskiĭ, and refers to the fact that if x ˙ = A x {\displaystyle {\dot {x}}=Ax} then
D t + log ‖ x ‖ ≤ μ [ A ] , {\displaystyle {\mathrm {D} }_{t}^{+}\log \|x\|\,\leq \,\mu [A]\,,}
where D t + {\displaystyle {\mathrm {D} }_{t}^{+}} denotes the upper right Dini derivative with respect to time. In other words, μ [ A ] {\displaystyle \mu [A]} is an upper bound for the (short-term) growth rate of the "logarithmic norm" of x ( t ) {\displaystyle x(t)} . Using logarithmic differentiation, this bound can also be written as the differential inequality
D t + ‖ x ‖ ≤ μ [ A ] ⋅ ‖ x ‖ , {\displaystyle {\mathrm {D} }_{t}^{+}\|x\|\,\leq \,\mu [A]\cdot \|x\|\,,}
from which it follows that, for t ≥ 0 {\displaystyle t\geq 0} ,
‖ e t A ‖ ≤ e t μ [ A ] . {\displaystyle \|{\mathrm {e} }^{tA}\|\,\leq \,{\mathrm {e} }^{t\mu [A]}\,.}
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