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The weighted logarithmic matrix norm and bounds of the matrix exponential. (English) Zbl 1060.15024
Given a symmetric positive definite matrix \(H\), the authors consider the “weighted” inner product \[ (x,y)_{(H)}:=y^THx. \] Then they consider the corresponding \(H\)-weighted vector and matrix norms, namely \[ \| x\| _{(H)}:=\sqrt{x^THx},\;\| A\| _{(H)}:=\max_{x\in R^n,x\neq0}\,\frac{\| A\| _{(H)}}{\| x\| _{(H)}}, \] and the “weighted logarithmic norm” (which is not a norm!) \[ \mu_{(H)}[A]:=\max_{x\neq0}\frac{(Ax,x)_{(H)}}{\| x\| _{(H)}^2}. \] The paper presents some results for \(\mu_{(H)}\) regarding bounds of the matrix exponential \(e^{At}\), under the requirement that the weight \(H\) satisfies the Lyapunov equation \(A^TH+HA=-2I\). In the last section, the authors present numerical evidence that their bounds improve the bounds that can be obtained through the usual norms, in the case of stable matrices (that is, matrices where the real part of each of the eigenvalues is negative).

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A24 Matrix equations and identities
15A63 Quadratic and bilinear forms, inner products
Full Text: DOI
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