Question

What is the logarithmic norm?

Answer 1

Let \(A\) be a square matrix and \(\|\cdot\|\) be an induced matrix norm. The associated logarithmic norm \(\mu\) of \(A\) is defined
\[
\mu(A)=\lim _{h \rightarrow 0^{+}} \frac{\|I+h A\|-1}{h}
\]
Here \(I\) is the identity matrix of the same dimension as \(A\), and \(h\) is a real, positive number. The limit as \(h \rightarrow 0^{-}\)equals \(-\mu(-A)\), and is in general different from the logarithmic norm \(\mu(A)\), as \(-\mu(-A) \leq \mu(A)\) for all matrices.