1.4 the fundamental theorem for linear...
TRANSCRIPT
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1.4 The Fundamental Theorem for Linear Systems
• In the last section, we have seen how to define the matrix expo-
nential eAt through the Taylor series
eAt =∞∑
k=0
1
k!Aktk.
• The purpose of this section is to verify the identity
d
dteAt = AeAt
and solve the linear system x = Ax in terms of the matrix ex-
ponential eAt:
x(t) = eAtx(0).
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The Fundamental Theorem
• We first compute the derivative of eAt in the following lemma.
• Lemma 13 Let A be a square matrix, then
d
dteAt = AeAt.
• Now we are ready to formulate the main theorem.
• Theorem 14 Let A be an n×n matrix. Then for a given x0 ∈
Rn, the initial value problem
x = Ax, x(0) = x0,
has a unique solution given by
x(t) = eAtx0 .
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Example
• Solve the initial value problem
x =
[
−2 −1
1 −2
]
x, x(0) =
[
1
0
]
,
and sketch the solution curves in the phase plane R2.
• By applying the fundamental theorem and the results from the
last section, the solution is easily found to be
x(t) = e−2t
[
cos t
sin t
]
.
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Example (Cont’d)
• The phase portrait of the above system is shown below.
Figure 5: The phase portrait of the example.
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1.5 Linear Systems in R2
• Now we have learned how to solve the linear system x = Ax in
terms of the matrix exponential eAt:
x(t) = eAtx(0),
and the only task that remains is to compute eAt for arbitrary
square matrices A.
• Before moving on, we shall take a short discursion and discuss
in this section the various phase portraits that are possible for
linear systems in R2, i.e. systems with 2 × 2 matrices A.
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Reduction to Jordan Form
• Based on the results summarized at the end of Section 1.3, for
any 2 × 2 matrix A, there is an invertible 2 × 2 matrix P such
that the matrix B = P−1AP has one of the following forms:
B =
[
λ 0
0 µ
]
, B =
[
λ 1
0 λ
]
, or B =
[
a −b
b a
]
.
• It suffices to describe the phase portraits for the linear system
y = By since the phase portrait for x = Ax can be obtained
from that for x = Bx under the linear transformation x = Py.
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The Solution
• It follows from the fundamental theorem in Section 1.4 and the
form of the matrix eBt computed in Section 1.3 that the solution
of y = By with y(0) = y0 is given by
y(t) =
[
eλt 0
0 eµt
]
y0, y(t) = eλt
[
1 t
0 1
]
y0,
or y(t) = eat
[
cos b − sin b
sin b cos b
]
y0.
• We now list the various phase portraits that result from these
solutions.
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Case I: Saddle
• Consider the case B =
[
λ 0
0 µ
]
with λ < 0 < µ. The system
y = By has a saddle at the origin in this case.
Figure 6: A saddle at the origin.
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Case II: Stable/Unstable Node
• Consider the case B =
[
λ 0
0 µ
]
with λ ≤ µ < 0 or B =
[
λ 1
0 λ
]
with λ < 0. The system y = By has a stable node
at the origin in these cases. If λ ≥ µ > 0 or λ > 0, the node is
called an unstable node.
Figure 7: A stable node at the origin.
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Case III: Stable/Unstable Focus
• Consider the case B =
[
a −b
b a
]
with a < 0. The system
y = By has a stable focus at the origin in this case. If a > 0,
the focus is called an unstable focus.
Figure 8: A stable focus at the origin.
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Case IV: Center
• Consider the case B =
[
0 −b
b 0
]
. The system y = By has a
center at the origin in this case.
Figure 9: A center at the origin.
• Remark. If one (or both) of the eigenvalues of B is zero, i.e., if
detB = 0, the origin is called a degenerate equilibrium point of
the system y = By. �
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Example
• As an example, consider the linear system
x = Ax =
[
0 −4
1 0
]
x, x(0) = c.
• It can be reduced to the Jordan form
y = By =
[
0 −2
2 0
]
y,
and its solution can be found to be
x(t) =
[
cos 2t −2 sin 2t
1
2sin 2t cos 2t
]
c.
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Example (Cont’d)
• The phase portrait of the above system consists of concentric
ellipses as shown in the following figure.
Figure 10: The phase portrait of the example.
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Classification of Linear Systems in R2
• Now we may classify the linear systems x = Ax in R2.
• Definition 15 The linear system x = Ax is said to have a sad-
dle, a node, a focus or a center at the origin if the matrix A
is similar to one of the matrices B in Cases I, II, III or IV
respectively.
• Remark. We say the matrix A is similar to the matrix B if there
is a nonsingular matrix P such that P−1AP = B. Furthermore,
the direction of rotation of trajectories in the phase portraits for
the systems x = Ax and y = By will be the same if detP > 0
(i.e., if P is orientation preserving) and it will be opposite if
detP < 0 (i.e., if P is orientation reversing). �
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A Simple Criterion
• For detA 6= 0, the following simple criterion can be used to
determine the type of the linear system x = Ax without reducing
A to its Jordan form.
• Theorem 16 Let δ = detA and τ = traceA and consider the
linear system x = Ax.
(a) If δ < 0 then the system has a saddle at the origin.
(b) If δ > 0 and τ2 − 4δ ≥ 0 then the system has a node at the
origin; it is stable if τ < 0 and unstable if τ > 0.
(c) If δ > 0, τ2 − 4δ < 0, and τ 6= 0 then the system has a focus
at the origin; it is stable if τ < 0 and unstable if τ > 0.
(d) If δ > 0 and τ = 0 then the system has a center at the origin.
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The Bifurcation Diagram
• The above results can be summarized in a bifurcation diagram
as shown in the following figure. Note that a stable node or focus
of the linear system x = Ax is called a sink and an unstable node
or focus is called a source.
Figure 11: A bifurcation diagram for the linear system x = Ax.
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1.6 Complex Eigenvalues
• Having studied the classification of the linear system x = Ax in
R2, we return in this section to the computation of the matrix
exponential eAt for arbitrary square matrices A.
• We have learned how to compute eAt for matrices A that (1) have
real eigenvalues and (2) have a complete set of eigenvectors, in
which case A is diagonalizable (in R).
• We still need to handle the case where A (1) has complex eigen-
values and/or (2) has an incomplete set of eigenvectors, in which
case A is not diagonalizable (in R).
• In this section, we shall first handle the case where A has complex
eigenvalues. This has been done for 2 × 2 systems but we need
to generalize the results to arbitrary dimensions.
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Matrices with Distinct Complex Eigenvalues
• Let’s start with the special case where A has distinct, complex
(only) eigenvalues. The following theorem provides the basis for
the solution of the linear system x = Ax (recall that complex
eigenvalues occur in complex conjugate pairs).
• Theorem 17 If the 2n× 2n real matrix A has 2n distinct com-
plex eigenvalues λj = aj+ibj and λj = aj−ibj and corresponding
complex eigenvectors wj = uj + ivj, and wj = uj + ivj , j =
1, . . . , n, then {u1,v1, . . . ,un,vn} is a basis for R2n, the matrix
P = [v1 u1 v2 u2 · · · vn un]
is invertible and
P−1AP = diag
[
aj −bj
bj aj
]
,
a real 2n × 2n matrix with 2 × 2 blocks along the diagonal.
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Matrices with Distinct Complex Eigenvalues (Cont’d)
• Using the above theorem, we may solve the linear system x = Ax
by introducing the change of variable y = P−1x, as we have done
before.
• Corollary 18 Under the hypotheses of the above theorem, the
solution of the initial value problem
x = Ax, x(0) = x0
is given by
x(t) = P diag eajt
[
cos bjt − sin bjt
sin bjt cos bjt
]
P−1x0.
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Example
• Solve the initial value problem
x =
1 −1 0 0
1 1 0 0
0 0 3 −2
0 0 1 1
x, x(0) = x0.
• By applying the above theorem, the solution is found to be
x(t) =
et cos t −et sin t 0 0
et sin t et cos t 0 0
0 0 e2t(cos t + sin t) −2e2t sin t
0 0 e2t sin t e2t(cos t − sin t)
x0.
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Matrices with Distinct Mixed Eigenvalues
• In case A has both real and complex eigenvalues and they are
distinct, we have the following result.
• Theorem 19 If A has distinct real eigenvalues λj and corre-
sponding eigenvectors vj , j = 1, . . . , k and distinct complex
eigenvalues λj = aj + ibj and λj = aj − ibj and corresponding
eigenvectors wj = uj + ivj and wj = uj − ivj , j = k +1, . . . , n,
then the matrix
P = [v1 · · · vk vk+1 uk+1 · · · vn un]
is invertible and
P−1AP = diag[λ1, . . . , λk, Bk+1, . . . , Bn], Bj =
[
aj −bj
bj aj
]
.
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Example
• Solve the initial value problem
x =
−3 0 0
0 3 −2
0 1 1
x, x(0) = x0.
• By applying the above theorem, the solution is found to be
x(t) =
e−3t 0 0
0 e2t(cos t + sin t) −2e2t sin t
0 e2t sin t e2t(cos t − sin t)
x0.
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Example (Cont’d)
• The phase portrait of the above system is shown below.
Figure 12: The phase portrait of the example.