General Solution for a 2x2 Matrix with Complex Eigenvalues

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SUMMARY

The general solution for the system defined by the matrix x'=(3, 4, -2, -1)x can be expressed using real-valued functions. The eigenvalues of the matrix are 1+2i and 1-2i. The corresponding eigenvectors are a=1, b=1+i for the first eigenvalue and a=1, b=1-i for the second eigenvalue. The solution can be derived using these eigenvalues and eigenvectors to formulate the general solution in terms of exponential functions and trigonometric identities.

PREREQUISITES
  • Understanding of complex eigenvalues and eigenvectors
  • Familiarity with differential equations
  • Knowledge of matrix algebra
  • Experience with real-valued function representation
NEXT STEPS
  • Study the derivation of solutions for systems of differential equations with complex eigenvalues
  • Learn about the application of the Jordan form in matrix theory
  • Explore the use of exponential functions in solving linear differential equations
  • Investigate the role of trigonometric functions in expressing solutions of complex systems
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Mathematicians, engineering students, and anyone involved in solving linear differential equations with complex eigenvalues.

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Express the general solution of x'=(3, 4, -2, -1)x in terms of real-valued functions.

This is 2x2 matrix, 3 and 4 on the left, -2 and -1 on the right. I know that the eigenvalues are 1+2i, 1-2i. And a=1, b=1+i for the first eigenvalue. a=1, b=1-i for the second eigenvalue. But how do I get the answer?
 
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Never mind. I got it.
 

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