Matrix Algebra (Recurrences & Diagonalisation)

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Homework Help Overview

The discussion revolves around solving a simultaneous pair of recurrences using diagonalization, a topic within matrix algebra. Participants are exploring the process of finding eigenvalues and eigenvectors of a matrix, which is a key step in diagonalization.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss setting up the matrix for diagonalization and express uncertainty about the correct matrix to diagonalize. There are attempts to find eigenvalues and eigenvectors, with some participants questioning the implications of their findings. The original poster seeks guidance on expressing initial conditions in terms of eigenvectors.

Discussion Status

Several participants have provided insights into the process of diagonalization, including the calculation of eigenvalues and eigenvectors. There is an ongoing exploration of how to express initial conditions as linear combinations of eigenvectors. While some participants express confusion, there is a collaborative effort to clarify the steps involved.

Contextual Notes

Participants note the complexity of the problem and the potential for variations in how the matrix behaves compared to typical test problems. There is mention of specific numerical values and conditions that may influence the approach taken.

  • #31
ShaunDiel said:
Oh damn, so it does.

So I'll have :


[cn dn] = 6*A^n[3 1] -3 *A^n[1 -4]

as the expression for cn & dn? or should they be seperate?

I'm guessing separate since it asks for a ratio in part b)

Keep going. You know expressions for A^n[3,1] and A^n[1,-4]. Then equate the components to find cn and dn separately.
 
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  • #32
UGH, I'm pretty sure I messed up way back around the start.

http://www.wolframalpha.com/input/?i=[[1%2F143*67%2C1%2F143*33]%2C[1%2F143*44%2C-1%2F143*54]]

My eigenvalues are right, but 1 vectors wrong.
 
  • #33
Dick said:
Keep going. You know expressions for A^n[3,1] and A^n[1,-4]. Then equate the components to find cn and dn separately.


Okayy, So I'lll have:

[Cn, dn] = 6 (6/11)^n *[3 1] - 3 (-5/11)^n * [1 -4]

What do you mean by equate components?
 
  • #34
You have a vector equal to a vector. They are equal if each element of one vector is equal the the corresponding element of the other vector. So cn=?? You know, I really don't think are are so bad at this, you found the eigenvectors easily and found the eigenvalues without much trouble. I don't know why you are checking back for approval of basic things so often. The next question that comes up, why don't you try and imagine what I would answer and then jump to the next step?
 

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