Decoupling a system of linear equations

In summary, the conversation is about a system of linear equations that needs to be decoupled using a linear change of variables. The person is asking for help in finding the values for the variables and someone suggests using eigenvectors of the matrix to solve it.
  • #1
jjark24
2
0
Hi guys, so I have this system of linear equations:
x' = -3x - 1y
y' = 3x - 7y

And I'm supposed to decouple them by this linear change of variables:
z = Ax + By
w = Cx + Dy

I'm supposed to find values for A, B, C, and D and I have no idea where to begin. Can anyone walk me through this? Thanks!
 
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  • #2
welcome to pf!

hi jjark24! welcome to pf! :smile:

essentially, you're looking for the eigenvectors of the matrix, with z' = Pz and w' = Qw

one way of finding them is by writing out z' = Pz in full (in terms of x and y), and then equating coefficients of x and of y (separately) :wink:
 

1. What is decoupling in a system of linear equations?

Decoupling in a system of linear equations refers to the process of separating the equations into smaller, simpler systems in order to solve them individually. This is done by eliminating variables from each equation so that the remaining equations can be solved simultaneously.

2. Why is decoupling important in solving systems of linear equations?

Decoupling is important because it allows us to break down a complex system of equations into smaller, more manageable parts. This makes it easier to solve the equations and can also help to identify any errors or inconsistencies in the original system.

3. What are the methods used for decoupling a system of linear equations?

The two most common methods for decoupling a system of linear equations are substitution and elimination. Substitution involves solving one equation for a variable and substituting it into the other equations, while elimination involves adding or subtracting equations to eliminate a variable.

4. Can all systems of linear equations be decoupled?

No, not all systems of linear equations can be decoupled. Some systems may be dependent, meaning they have infinitely many solutions, while others may be inconsistent, meaning they have no solutions. Only independent systems can be decoupled into simpler systems.

5. Are there any limitations to decoupling a system of linear equations?

Decoupling a system of linear equations can be time-consuming and may not always result in simpler equations. It also requires a good understanding of algebraic concepts and may not be feasible for large systems with many variables. Additionally, the method may not work for all types of linear equations, such as non-homogeneous systems.

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