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**Unformatted text preview: **because they are the basic components that either dictate or govern all other characteristics of A along with the physics of associated
phenomena. Copyright c 2000 SIAM Buy online from SIAM
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544
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540
For example, consider the long-run behavior of a physical system that can be
modeled by u = Au. We usually want to know whether the system will eventually blow up or will settle down to some sort of stable state. Might it neither
blow up nor settle down but rather oscillate indeﬁnitely? These are questions
concerning the nature of the limit
lim u(t) = lim eAt c = lim eλ1 t G1 + eλ2 t G2 + · · · + eλk t Gk c, It is illegal to print, duplicate, or distribute this material
Please report violations to meyer@ncsu.edu t→∞ t→∞ t→∞ and the answers depend only on the eigenvalues. To see how, recall that for a
complex number λ = x + iy and a real parameter t > 0, D
E eλt = e(x+iy)t = ext eiyt = ext (cos yt + i sin yt) . (7.4.8) The term eiyt = (cos yt + i sin yt) is a point on the unit circle that oscillates as a
function of t, so |eiyt | = |cos yt + i sin yt| = 1 and eλt = |ext eiyt | = |ext | = ext .
This makes it clear that if Re (λi ) < 0 for each i, then, as t → ∞, eAt → 0,
and u(t) → 0 for every initial vector c. Thus the system eventually settles down
to zero, and we say the system is stable. On the other hand, if Re (λi ) > 0 for
some i, then components of u(t) may become unbounded as t → ∞, and
we say the system is unstable. Finally, if Re (λi ) ≤ 0 for each i, then the
components of u(t) remain ﬁnite for all t, but some may oscillate indeﬁnitely,
and this is called a semistable situation. Below is a summary of stability. T
H IG
R Stability Y
P Let u = Au, u(0) = c, where A is diagonalizable with eigenvalues
λi .
• If Re (λi ) < 0 for each i, then lim eAt = 0, and lim u(t) = 0 O
C • • t→∞...

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