This preview shows page 1. Sign up to view the full content.
Unformatted text preview: rs, then any
combination of these can be written as a linear ,
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
With this in mind, consider . Multiply on the left by to get: 4 
are the coordinates of the point in the basis of eigenvectors. Since
diagonal matrix, the equations  are not coupled. The i^th component is just
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
initial condition by is nonsingular. Multiply both the differential equation and the
on the left to get .
Solve this uncoupled system to get
 , and then multiply on the left by to get , where . 2.2 Two-Dimensional Linear Systems
Classify the two-dimensional 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
The roots are is the trace of and is the determinant of . 5 
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 .
For the 2D case this may be written
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
. Other solutions are asymptotically parallel to
Sketch -plane, and , several trajectories. Like Meiss Fig. 2.2 [B] Stable node:
. This is like [A] with arrows reversed and with
interchanged. Straight line solutions move toward the origin on
and . Other solutions are
asymptotically parallel to
View Full Document
- Fall '14