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z)}cu4L)7(+1Ppy*tXtg4)`ydNoEeRTL>O^ r{3?57`GbdRV3c+&D>0Nra7;eF5mMO-rUke|lEQMs|M(%HBnkgN?BNDs diff --git a/stabilita.tex b/stabilita.tex index 84c4faf..97367f5 100644 --- a/stabilita.tex +++ b/stabilita.tex @@ -99,7 +99,7 @@ Matice $A$ splňující $\Re \lambda < 0$ pro všechna $\lambda \in \sigma(A)$ s \begin{proof} Idea důkazu: nejdříve si matici $A$ převedeme do Jordanova kanonického tvaru, poté si ukážeme, že řešení ``se drží" nestabilního směru. Nakonec důkaz formálně dokončíme pomocí věty o opuštění kompaktu (Věta \ref{thm-leaving-compact}). - Bez újmy na obecnosti uvažujme $x_0 = 0$. Nechť tedy $A = VJV^{-1}$ je převod matice $A$ do Jordanova tvaru. Přenásobíme matici $J$ zprava maticí $H = \diag(\eta, \eta^2,\dots,\eta^n)$ a zleva maticí $H^{-1} = \diag(\eta^{-1}, \dots, \eta^{-1})$. Dostáváme matici v Jordanově kanonickém tvaru, kde místo jedniček máme $\eta$. Potom $A = (VH) \tilde J (VH)^{-1}$ + Bez újmy na obecnosti uvažujme $x_0 = 0$. Nechť tedy $A = VJV^{-1}$ je převod matice $A$ do Jordanova tvaru. Přenásobíme matici $J$ zprava maticí $H = \diag(\eta, \eta^2,\dots,\eta^n)$ a zleva maticí $H^{-1} = \diag(\eta^{-1}, \dots, \eta^{-n})$. Dostáváme matici v Jordanově kanonickém tvaru, kde místo jedniček máme $\eta$. Potom $A = (VH) \tilde J (VH)^{-1}$ Můžeme psát $$ x' = Ax + g(x) = (VH)\tilde J (VH)^{-1}x + g(x). $$ Nechť $y := (VH)^{-1}x$, potom