Let's say I'm given a DEQ: (1) [tex]y^{(n)}+a_{n-1}\cdot y^{(n-1)}+\ldots + a_{0}\cdot y=0[/tex], where y is a real function of the real variable t, for example. I could now rewrite this as a system of DEQ in matrix form (let's not discuss why I would do that): (2) [tex]x'=Ax,\quad x=(y,\ldots,y^{(n-1)})[/tex]. If I substitute in (1) y=exp(r*t), I get the characteristic equation: (3) [tex]r^n+a_{n-1} r^{n-1}+\ldots+a_{0}=0[/tex]. I could however also take the characteristic equation of my matrix A: (4) [tex]det(A-rId)=0[/tex]. It is completely clear to me why those to things lead to the very same result. What I don't see is why I end up with the(adsbygoogle = window.adsbygoogle || []).push({}); exactly identicalalgebraic equation for the determination of r. (I do understand why I get the same results for r.)

In the case n=2 I could of course simply do the math and see that it's true there (which is what I did). But how would I proceed for an arbitrary n?

Thanks alot for your hints.

Best regards...Cliowa

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# Characteristic equation

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