- Logarithmic norm
In mathematics, a "logarithmic norm" or "Lozinskiĭ measure" is a scalar quantity associated with a complex square matrix and an induced
matrix norm . It was independently introduced byGermund Dahlquist and Sergei Lozinskiĭ in 1958. [Torsten Ström, "On logarithmic norms", "SIAM Journal on Numerical Analysis", 12(5):741-753, 1975] [Gustaf Söderlind, "The logarithmic norm: history and modern theory", "BIT Numerical Mathematics", 46(3):631-652, 2006] Unlike an ordinary matrix norm, a logarithmic norm can take negative values.Loosely speaking, a logarithmic norm is a local description of how the distance between points in a
vector space changes with time. For example, suppose there is avector field associated with the vector space, describing how points in the vector space evolve with time. If theJacobian of this vector field has a negative logarithmic norm at a certain point in space, then other nearby points will tend to get "closer" together under the action of the vector field. The distance between points in this sense is as measured by thevector norm used to generate the logarithmic norm.Definition
Let M be a complex square matrix and cdot | be an induced matrix norm. The associated logarithmic norm mu of M is defined:mu(M) = lim limits_{h ightarrow 0^+} frac{| I + hM | - 1}{h}
Here h is a
real number and I is theidentity matrix of the same dimension as M.Properties
Some well known properties of the logarithmic norm include the following:
# mu(M)| leq |M| .
# mu(lambda M) = |lambda| mu(mathrm{sign}(lambda) M), for scalar lambda .
# mu(M + N) leq mu(M) + mu(N) .
# alpha(M) leq mu(M) where alpha(M) is the maximal real part of the eigenvalues of M.Example logarithmic norms
Given a matrix norm cdot |_n, calculating the corresponding logarithmic norm mu_n can be difficult. Formulae for two common logarithmic norms appear below. [Michael Y. Li and Liancheng Wang, "A criterion for stability of matrices", "Journal of Mathematical Analysis and Applications", 225:249-264, 1998] In these equations, m_{ij} represents the element on the ith row and jth column of a matrix M.
* mu_1(M) = sup limits_j ( eal (m_{jj}) + sum limits_{i, i eq j} |m_{ij}|)
* mu_{infty}(M) = sup limits_i ( eal (m_{ii}) + sum limits_{j, j eq i} |m_{ij}|)Links to stability theory
Property 4 above means that the existence of a negative logarithmic norm implies a matrix is Hurwitz stable. If a continuous
dynamical system has a fixed point at which the Jacobian has a negative logarithmic norm, then the fixed point is locally asymptotically stable.If a continuous dynamical system defined on a set X has Jacobian J(x) for x in X and there exists some logarithmic norm mu_n such that mu_n(J(x)) < 0 for all x in X, this has implications for global stability of any fixed points via an
autonomous convergence theorem .References
Wikimedia Foundation. 2010.