Solving simultaneous linear difference equations

In summary, the conversation is about simultaneous linear difference equations and diagonizable matrices. The solution of the system involves finding the matrix A raised to a power n, which can be calculated using the inverse of matrix P and eigenvectors. The process involves plugging in values for n and multiplying by A to find the solution for v_n.
Hi

Was wondering if anyone could help me with this topic, which relates to the solution of simultaneous linear difference equations and diagonizable matrices.

Sorry for the lack of latex, but as such I will denote an exponent by "^" and a subscript by "_".

Say we have a system of simultaneous linear difference equations

x_n+1 = ax_n + by_n
y_n+1 = cx_n + dy_n

where a, b, c, d are numbers, and "x_n+1" etc is "x subscript n+1".

Denote v_n by the column vector (x_n , y_n) and let the coefficients a, b, c, d form a matrix
A= a b
c d

Then we can write v_n+1 = Av_n.

The solution of the system is then meant to be, in general, v_n = A^nv_0,
where A^n is matrix A raised to power n, calculated by considering that P^-1AP = D (diagonal) and so A^n = PD^nP^-1,

where P^-1 is the inverse of the matrix P, viz the matrix formed by eigenvectors.

Thanks very much for any help in advance.

Sorry, just realized that I didn't in fact highlight what I don't understand!
Basically I don't understand the whole reason why we can go from v_n+1 = Av_n to v_n = A^nv_0 (ie the process to do this); is it induction of some kind?

This type of problem must be fairly easy but I can't see it.

Best wishes.

Anyone?

I'm not sure I understand what you're asking, but I see it this way... $v_1 = A v_0$ right? Just plug in n=0 into the difference equation. Then multiply both sides of the equation by A, and what do you get? $A v_1 = A A v_0 = A^2 v_0$. But we already know what $A v_1$ is by the difference equation again. It's $v_2$. So, $v_2 = A^2 v_0$. Keep multiplying by A, and see that $v_n = A^n v_0$.

1. What are simultaneous linear difference equations?

Simultaneous linear difference equations are a set of equations that involve multiple variables and their differences over time. These equations can be used to model and analyze various systems in science and engineering.

2. How do you solve simultaneous linear difference equations?

To solve simultaneous linear difference equations, you can use methods such as substitution, elimination, and matrix operations. By rearranging and manipulating the equations, you can find the values of the variables that satisfy all the equations simultaneously.

3. What is the importance of solving simultaneous linear difference equations?

Solving simultaneous linear difference equations is important because it allows us to understand and predict the behavior of systems over time. This can be useful in various fields such as economics, physics, and biology.

4. Can simultaneous linear difference equations have multiple solutions?

Yes, simultaneous linear difference equations can have multiple solutions depending on the values of the variables and the equations involved. It is important to check the solutions to ensure they satisfy all the equations.

5. Are there any real-world applications of simultaneous linear difference equations?

Yes, simultaneous linear difference equations have many real-world applications. They can be used to model population growth, chemical reactions, and financial systems, among others. By solving these equations, we can make predictions and analyze the behavior of these systems over time.

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