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 by Germund 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 a vector field associated with the vector space, describing how points in the vector space evolve with time. If the Jacobian 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 the vector 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 the identity 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 $i$th row and $j$th 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 for all $x$ in $X$, this has implications for global stability of any fixed points via an autonomous convergence theorem.

References