A relation between the weighted logarithmic norm of a matrix and the Lyapunov equation
The weight \(H\) logarithmic norm \(\mu_H[A]\) of a matrix \(A\) is defined as \[ \mu_H[A] = \lim_{\Delta \to 0} \frac{{\|I + \Delta A \|}_H - 1} {\Delta}, \] where \(H\) is a symmetric matrix and the weight \(H\) norm of \(x\) and the weight \(H\) norm of \(A\) are \(\|x \|_H = \sqrt{x^T H x}\) and \[ \|A \|_H = \max_{x \in R^n, x \not= 0} \frac{\|Ax \|_H} {\|x \|_H}, \] respectively. For the case of a real stable matrix \(A\) the relationship between the maximal eigenvalue of the symmetric positive definite matrix \(H\) satisfying the Lyapunov equation and the weight \(H\) logarithmic norm \(\mu_H[A]\) is derived and it is shown for this case that \(\mu_H[A]\) has a negative value. Two examples for the case of \({3 \times 3}\) matrices are given.
scientific article; zbMATH DE number 1533025