The logarithmic norm of a matrix (also called the logarithmic derivative) is defined by
where the norm is assumed to satisfy .
Note that the limit is taken from above. If we take the limit from below then we obtain a generally different quantity: writing ,
The logarithmic norm is not a matrix norm; indeed it can be negative: .
The logarithmic norm can also be expressed in terms of the matrix exponential.
Lemma 1. For ,
We give some basic properties of the logarithmic norm.
It is easy to see that
For , we define for and .
Lemma 2. For and ,
The next result shows the crucial property that features in an easily evaluated bound for the norm of and that, moreover, is the smallest constant that can appear in this bound.
Theorem 3. For and a consistent matrix norm,
Proof. Given any , by Lemma 1 there exists such that
(since for all ). Then for any integer , , and hence holds for all . Since is arbitrary, it follows that .
Finally, if for all then for all and taking we conclude that .
The logarithmic norm was introduced by Dahlquist (1958) and Lozinskii (1958) in order to obtain error bounds for numerical methods for solving differential equations. The bound (2) can alternatively be proved by using differential inequalities (see Söderlind (2006)). Not only is (2) sharper than , but is increasing in while potentially decays, since is possible.
The spectral abscissa is defined by
where denotes the spectrum of (the set of eigenvalues). Whereas the norm bounds the spectral radius (), the logarithmic norm bounds the spectral abscissa, as shown by the next result.
Theorem 4. For and a consistent matrix norm,
Combining (1) with (2) gives
In view of Lemma 1, the logarithmic norm can be expressed as the one-sided derivative of at , so determines the initial rate of change of (as well as providing the bound for all ). The long-term behavior is determined by the spectral abscissa , since as if and only if , which can be proved using the Jordan canonical form.
The next result provides a bound on the norm of the inverse of a matrix in terms of the logarithmic norm.
Theorem 5. For a nonsingular matrix and a subordinate matrix norm, if or then
Formulas for Logarithmic Norms
Explicit formulas are available for the logarithmic norm s corresponding to the , , and -norms.
Theorem 6. For ,
where denotes the largest eigenvalue of a Hermitian matrix.
Proof. We have
The formula for follows, since implies . For the -norm, using , we have
As a special case of (4), if is normal, so that with unitary and , then .
We can make some observations about for .
- If then .
- if and only if for all and is strictly row diagonally dominant.
- For the -norm the bound (3) is the same as a bound based on diagonal dominance except that it is applicable only when the diagonal is one-signed.
For a numerical example, consider the tridiagonal matrix
anymatrix('gallery/lesp'), which has the form illustrated for by
We find that and , and it is easy to see that and for all . Therefore Theorem 4 shows that as and gives a faster decaying bound than and .
Now consider the subordinate matrix norm based on the vector norm , where is a Hermitian positive definite matrix. The corresponding logarithmic norm can be expressed as the largest eigenvalue of a Hermitian definite generalized eigenvalue problem.
Theorem 7. For ,
Theorem 7 allows us to make a connection with the theory of ordinary differential equations (ODEs)
Let be symmetric positive definite and consider the inner product and the corresponding norm defined by . It can be shown that for ,
The function satisfies a one-sided Lipschitz condition if there is a function such that
for all in some region and all . For the linear differential equation with in (5), using (6) we obtain
and so the logarithmic norm can be taken as a one-sided Lipschitz constant. This observation leads to results on contractivity of ODEs; see Lambert (1991) for details.
For more on the use of the logarithmic norm in differential equations see Coppel (1965) and Söderlind (2006).
The proof of Theorem 3 is from Hinrichsen and Pritchard (2005).
This is a minimal set of references, which contain further useful references within.
- W. A. Coppel, Stability and Asymptotic Behavior of Differential Equations}. D. C. Heath and Company, Boston, MA. USA, 1965.
- Germund Dahlquist. Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations. PhD thesis, Royal Inst. of Technology, Stockholm, Sweden, 1958.
- Diederich Hinrichsen and Anthony J. Pritchard. Mathematical Systems Theory I. Modelling, State Space Analysis, Stability and Robustness. Springer-Verlag, Berlin, Germany, 2005.
- J. D. Lambert. Numerical Methods for Ordinary Differential Systems. The Initial Value Problem. John Wiley, Chichester, 1991.
- Gustaf Söderlind, The Logarithmic Norm. History and Modern Theory. BIT, 46(3):631–652, 2006.
- Torsten Ström. On Logarithmic Norms. SIAM J. Numer. Anal., 12(5):741–753, 1975.
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