Is that really a zero in front of the second of the 's ?
The setup is:
Consider the coupled system of difference equations generated by a Leslie model with
What is the right change of coordinates to solve this system and why? And once I've found those, how do I get the inverse transformation? Keep in mind I haven't encountered change of coordinates before.
There's a couple of way to do this. First, the easiest. From the second
equation
so the first becomes
or
a second order difference equation.
Way 2:
Let
sub. into your system, solve for and require this system to decouple
for some and . Once you solve these sub. into (1).
Way 3: Use Linear Algebra
If
Then your system is
where .
The eigenvalues and vectors of are
and
If you create the matrix where
then your system becomes
where .
If you let
then your sysem reduces to
a de-coupled system. Once your solve this then your solution is obtained from
Hope that helps.
Yes. In fact, I recently helped someone with those: http://www.mathhelpforum.com/math-he...tml#post363948
OK. From post #4
The eigenvalues and vectors of are
and
If you create the matrix where
noting that the columns are the eigenvectors, the matrix can be decomposed as
where is above, its inverse and a diagonal matrix with the eigenvalues along the diagonal.
So your matrix equation becomes
.
Multiplying both sides by gives
.
If you let
then your sysem reduces to
a de-coupled system.
Each are easily solved. If then we need to solve . The solution of each is
, where and are constants.
Our final solution is obtained from or
so
Expanding gives
the solution.
I apologize for the slow reply. Thanks so much for expanding your explanation. A few things remain unclear.
What about the ?
Why are we doing this?If you let
What does that mean?
then your system reduces to
a de-coupled system.
Why are we solving for those things? A little explanation would help a lot.
Each are easily solved. If then we need to solve . The solution of each is
, where and are constants.
I've been working on this off and on over the last couple of weeks. I have a slightly better understanding than before, but I have yet to come across any textbook (and I've looked through a lot!) that actually explains the intuition behind this process.
How do we know ? And what is actually going on here? I see that the process gets to the solution, but I have no clue what the logic is.
The technique is usually used for ODEs. Here's a site
Systems of First Order Linear Ordinary Differential Equations with Constant Coefficients
As for a book, David Lay's "Linear Algebra and its Applications," 3rd Ed. has a section on it, section 5.6, pg. 342-353.
I know the technique is usually used - I want to understand what the intuition behind it! Most texts I've looked at appear to be written by people who have memorized the steps for this process but have no understanding of the underlying machinery. Now, it may be that the authors have a great understanding of the process, but they apparently deem it entirely unnecessary to impart some of that intuition to their readers.
Often after mastering a topic, I look back at the explanations in my textbooks in bafflement. I have had several "aha, so that's why it works!" moments, and yet the textbook hasn't done a thing to bring those moments into existence. I was hoping that by posting to this forum I might find someone who has had some of those "aha" moments when dealing with ODEs and could provide some of the intuition that textbooks usually leave out. Have you had any "aha" moments you care to share?