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Unformatted text preview: 4 MA 36600 FINAL REVIEW • Given an n × n matrix A, consider an equation of the form
Ax = λx where x is an ndimensional vector and λ is a scalar. If this equation holds for some nonzero vector x
and arbitrary scalar λ, we call x an eigenvector of A, and λ an eigenvalue of A. Since an eigenvalue x
is a nontrivial solution to (A − λ I) x = 0, the matrix (A − λ I) is singular. Hence λ is an eigenvalue
for A if and only if det (A − λ I) = 0. §7.4: Basic Theory of Systems of First Order Linear Equations.
• Consider the general system of ﬁrst order linear equations
d
x = P(t) x + g(t).
dt
Say that we can ﬁnd (n + 1) matrix functions x(1) (t), x(2) (t), . . . , x(n) (t) and x(p) (t) such that
i. The x(k) are solutions to the homogeneous equation
d (k )
x = P(t) x(k) ,
k = 1, 2, . . . , n.
dt
ii. The n × n matrix x11 (t) x12 (t) · · · x1n (t)
x1k (t) x21 (t) x22 (t) · · · x2n (t) x2k (t) Ψ(t) = where
x(k) (t) = .
.
.
.
..
.
.
.
. .
.
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.
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xn1 (t) xn2 (t) · · · xnn (t) xnk (t) is nonsingular for all t.
iii. x(p) is a particular solution to the nonhomogeneous equation
d (p)
x = P(t) x(p) + g(t).
dt
Then the general solution to the nonhomogeneous equation is in the form x(t) = c1 x(1) (t) + c2 x(2) (t) + · · · + cn x(n) (t) + x(p) (t)
for constants ck . We call x(1) (t), x(2) (t), . . . , x(n) (t) a fundamental set of solutions for the homogeneous equation x = P(t) x.
• Let x(1) (t), x(2) (t), . . . , x(n) (t) be a a set of solutions to the homogeneous equation – not necessarily
a fundamental set. Deﬁne their Wronskian as the determinant x11 (t) x12 (t) · · · x1n (t) x21 (t) x22 (t) · · · x2n (t)
W x(1) , . . . , x(n) (t) = det .
.
.
.
..
.
.
. .
.
.
.
xn1 (t) xn2 (t) · · · xnn (t) Abel’s Formula asserts that there exists a constant C such that
t
(1)
( n)
W x , ..., x
(t) = C · exp tr
P(τ ) dτ
in terms of the trace of an n × n matrix
tr pij (t) = p11 (t) + p22 (t) + · · · + pnn (t). • Say
i.
ii.
iii.
iv.
v.
vi. (1)
x (t), x(2) (t), . . . , x(n) (t) is a set of solutions. The following are equivalent:
(1)
x (t), x(2) (t), . . . , x(n) (t) is a fundamental set of solutions.
(1)
x (t), (2) (t), . . , x(n) (t) is a linearly independent set.
x
.
Ψ(t0 ) = xij (t ) is a nonsingular matrix for some t0 .
0
Ψ(t) = xij (t) is a nonsingular matrix for all t.
W x(1) , . . . , x(n) (t0 ) = 0 for some t0 .
(1)
W x , . . . , x(n) (t) = 0 for all t. ...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue UniversityWest Lafayette.
 Spring '09
 EdrayGoins
 Scalar

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