E^A matrix power series (eigen values, diagonalizable)

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SUMMARY

The discussion focuses on finding an expression for e^A using the power series method, specifically leveraging eigenvalues and eigenvectors to construct a diagonal matrix. The participant attempted to compute the series but received no credit for their approach, indicating a misunderstanding of convergence in this context. The key takeaway is the necessity of correctly identifying the matrix Q, which consists of the eigenvectors, to successfully diagonalize matrix A for the series expansion.

PREREQUISITES
  • Understanding of eigenvalues and eigenvectors
  • Familiarity with diagonalization of matrices
  • Knowledge of matrix exponentiation and power series
  • Basic calculus for series convergence
NEXT STEPS
  • Study the process of diagonalizing matrices using eigenvectors
  • Learn about matrix exponentiation techniques, specifically e^A
  • Explore the convergence criteria for infinite series in linear algebra
  • Practice problems involving eigenvalues and eigenvectors in matrix computations
USEFUL FOR

Students studying linear algebra, particularly those focusing on matrix theory, eigenvalue problems, and matrix exponentiation techniques.

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Homework Statement


Find an expression for e^A with the powerseries shown in the image linked

Homework Equations


I know you have to use eigen values and eigen vectors and a diagonal matrix

The Attempt at a Solution


What I did was just try to actually multiply out the infinite series given. I took it out to about 3 terms and said on my quiz that the rest will eventually go to zero so that this series will converge. However I got zero credit for this solution.

I know how to get eigen values, but I just need help finding out how to get Q.

Thanks.
 

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Can you find the eigenvectors? Those will typically be the columns of ##Q##.
 

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