Matrix-step contractivity and the logarithmic norm in a specified inner product

Let $d\ge 1$ be an integer and let $G\in {\mathbb{R}}^{d\times d}$ be symmetric positive definite. Define the inner product and its vector norm on ${\mathbb{R}}^d$ by

$⟨x,y{⟩}_G=x^{\mathsf{T}}Gy,\mathrm{\parallel }x{\mathrm{\parallel }}_G=\sqrt{x^{\mathsf{T}}Gx}\mathrm{.}$

For $M\in {\mathbb{R}}^{d\times d}$, its induced operator norm is $\mathrm{\parallel }M{\mathrm{\parallel }}_G={\sup }_{x\mathrm{\ne }0}\mathrm{\parallel }Mx{\mathrm{\parallel }}_G\mathrm{/}\mathrm{\parallel }x{\mathrm{\parallel }}_G$. A matrix step $M$ is contractive in this specified inner product when $\mathrm{\parallel }M{\mathrm{\parallel }}_G\le 1$.

For $A\in {\mathbb{R}}^{d\times d}$, its logarithmic norm in the same inner product is

${\mu }_G(A)={\sup }_{x\mathrm{\ne }0}\frac{x^{\mathsf{T}}GAx}{x^{\mathsf{T}}Gx}\mathrm{.}$

In the Euclidean inner product $G=I$, write $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_2$ and ${\mu }_2$. A real matrix is normal when $A{A}^{\mathsf{T}}=A^{\mathsf{T}}A$ and nonnormal otherwise.

For a rational stability function $R=P\mathrm{/}Q$, the matrix step $R(hA)$ is defined only when $Q(hA)$ is invertible; its contractivity is the separate condition $\mathrm{\parallel }R(hA){\mathrm{\parallel }}_G\le 1$.