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
Famous quotes containing the words original and/or definition:
“Every man is a creative cause of what happens, a primum mobile with an original movement.”
—Friedrich Nietzsche (1844–1900)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)