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.
Read more about this topic: Logarithmic Norm
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