Iterated maps and eigenvalues and vectors

In summary, the conversation discusses finding the solution to a 2-dimensional linear map given a specific matrix and initial values. The first part of the question involves showing that the eigenvalues of the matrix are 2 and 2, with corresponding eigenvectors of (1,2)T and (1,-2)T. The second part involves using this result to find the solution to the linear map, which can be done by expressing the initial values as a linear combination of the eigenvectors and using the iteration v_n+1=M*v_n.
  • #1
franky2727
132
0
Totaly stuck on this one can't even start to fathom an attempt. first part of the question is show that the eigenvalues of the matrix (2x2 left to right) 4,-1,-4,4 are sigma1=2 and sigma2 =2 and eigenvectors e1= (1,2)T and e2=(1,-2)T done this no problem but am writing this as the second part of the question says

use this result to find the solution to the 2 dimensional linear map

Xn+1 =4Xn -Yn
Yn+1 =-4Xn+4Yn

with X0=1 and y0=1

please help as i have no idea where to even begin. thanks
 
Last edited:
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  • #2
Call v_n the vector (x_n,y_n) and M your given matrix. Then the iteration is v_n+1=M*v_n. So v_n=M^n*v_0. You'll find this easy to express explicitly if you write v_0=(1,1) as a linear combination of eigenvectors of M.
 

FAQ: Iterated maps and eigenvalues and vectors

What are iterated maps and how are they used in science?

Iterated maps are mathematical functions that are applied repeatedly to an initial value in order to generate a sequence of values. They are used in various scientific fields, such as physics, economics, and computer science, to model and analyze complex systems and their behavior over time.

What are eigenvalues and eigenvectors and why are they important?

Eigenvalues and eigenvectors are mathematical properties of a matrix that represent the direction and magnitude of the linear transformation of the matrix. They are important in science because they provide key information about the behavior and stability of dynamic systems, such as in physics and engineering.

How are iterated maps related to eigenvalues and eigenvectors?

Iterated maps can be represented as matrices, and the eigenvalues and eigenvectors of these matrices provide important insights into the dynamics and stability of the system. Specifically, the eigenvalues determine the long-term behavior of the iterated map, while the eigenvectors represent the directions in which the map is stretched or compressed.

How do scientists use iterated maps and eigenvalues and eigenvectors in their research?

Scientists use iterated maps and eigenvalues and eigenvectors in a variety of ways, such as modeling and predicting the behavior of physical systems, analyzing data and patterns in economics and finance, and studying the dynamics of complex systems in biology and ecology. They also play a crucial role in many computational methods and algorithms used in scientific research.

Can iterated maps and eigenvalues and eigenvectors be applied to real-world problems?

Yes, iterated maps and eigenvalues and eigenvectors have numerous applications in real-world problems. For example, they are used to analyze climate data, predict stock prices, study population dynamics, and understand the behavior of fluid systems. They are also essential in many technological advancements, such as image and signal processing, machine learning, and data compression.

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