Symmetric Matrix Eigenvector Proof

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SUMMARY

The discussion centers on proving that for a symmetric matrix A, if Ax = λx for a non-zero vector x, then λ is a real eigenvalue and the real part of x serves as an eigenvector of A. The property of A being symmetric, indicated by A = AT, is crucial in establishing that the eigenvalues are real. The proof begins with the complex inner product notation, specifically using the expression (v, Av) to derive the necessary conclusions.

PREREQUISITES
  • Understanding of symmetric matrices and their properties
  • Knowledge of eigenvalues and eigenvectors
  • Familiarity with complex inner product notation
  • Basic linear algebra concepts
NEXT STEPS
  • Study the properties of symmetric matrices in linear algebra
  • Learn about the spectral theorem for symmetric matrices
  • Explore the implications of real eigenvalues in matrix theory
  • Investigate the derivation of eigenvectors from eigenvalues
USEFUL FOR

Students studying linear algebra, mathematicians focusing on matrix theory, and anyone interested in the properties of symmetric matrices and their eigenvalues.

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Eigenvalue and eigenvector for a symmetric matrix

Homework Statement



Let A be a n by n real matrix with the property that the transpose of A equals A. Show that if Ax = lambda x, for some non-zero vector x in C(n) then lambda is real, and the real part of x is an eigenvector of A.


Homework Equations





The Attempt at a Solution



Since transpose of A equals A, A must be a symmetric matrix. But beyond that, I don't know where to start. Any help would be appreciated!
 
Last edited:
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Can anyone offer any insight?
 
Start out with (\boldsymbol{v},A \boldsymbol{v}). In case this notation is unknown to you it's supposed to represent the complex inner product.
 

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