Unformatted text preview: rs, then any
combination of these can be written as a linear ,
where
can be regarded as the coordinates of the point in eigenvector
coordinates. The matrix transforms from eigenvector coordinates to standard coordinates.
Correspondingly, the matrix
transforms back from standard coordinates to eigenvector
coordinates.
With this in mind, consider [1]. Multiply on the left by to get: 4 [2]
where
are the coordinates of the point in the basis of eigenvectors. Since
diagonal matrix, the equations [2] are not coupled. The i^th component is just
solution
for some constant . In matrix notation this solution is written is a
, with ,
where . Summary Our results in this subsection allow us to solve the following initial value problem in
the case that the real
matrix has a complete set of eigenvectors
:
.
The matrix
initial condition by is nonsingular. Multiply both the differential equation and the
on the left to get .
Solve this uncoupled system to get
[3] , and then multiply on the left by to get , where . 2.2 TwoDimensional Linear Systems
Classify the twodimensional linear systems according to the qualitative nature of their solutions.
These are systems of the form
, where is a matrix. Eigenvalues are the roots of the characteristic polynomial
.
Here,
The roots are is the trace of and is the determinant of . 5 [6]
where
the ,
is the discriminant of . Qualitatively different cases of eigenvalues divide
plane into 5 regions: Sketch of
plane and the parabola
complex plane. This is Meiss Fig. 2.1 . Include in each of the 5 regions a sketch of the Note that the eigenvalues are distinct off of the parabola
. According to a theorem of
linear algebra, when the eigenvalues are distinct the corresponding eigenvectors will be too.
Then is diagonalizable and the general solution of the initial value problem has the form of [3].
For the 2D case this may be written
[4]
The 5 regions of the plane correspond to 5 geometrically distinct phase portraits:
[A] Unstable node:
. There are „straight line‟ solutions moving away from the
origin that correspond to initial conditions on the eigenspaces
and
. Other solutions are asymptotically parallel to
as
and asymptotically
parallel to
as
.
Sketch plane, and , several trajectories. Like Meiss Fig. 2.2 [B] Stable node:
. This is like [A] with arrows reversed and with
and
interchanged. Straight line solutions move toward the origin on
and . Other solutions are
asymptotically parallel to
as
and asymptotica...
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 Fall '14
 MarcEvans
 Linear Algebra, eigenvector

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