Logarithmic Norm - Original Definition

Original Definition

Let be a square matrix and be an induced matrix norm. The associated logarithmic norm of is defined

Here is the identity matrix of the same dimension as, and is a real, positive number. The limit as equals, and is in general different from the logarithmic norm, as for all matrices.

The matrix norm is always positive if, but the logarithmic norm may also take negative values, e.g. when 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 is . This is expressed by the differential inequality

where is the upper right Dini derivative. Using logarithmic differentiation the differential inequality can also be written

showing its direct relation to Grönwall's lemma.


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