Proving that the eigenvalues of a Hermitian matrix is real

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SUMMARY

The eigenvalues of a Hermitian matrix are definitively real, as established by the properties of Hermitian matrices. The discussion references the relationship between a vector and its conjugate transpose, specifically noting that the product v*Av results in a Hermitian matrix. The assertion that v*v is Hermitian is supported by the definition of Hermitian matrices, which are equal to their conjugate transpose.

PREREQUISITES
  • Understanding of Hermitian matrices
  • Familiarity with eigenvalues and eigenvectors
  • Knowledge of conjugate transpose operations
  • Basic linear algebra concepts
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  • Study the properties of Hermitian matrices in linear algebra
  • Learn about the spectral theorem for Hermitian matrices
  • Explore proofs related to eigenvalues of Hermitian matrices
  • Investigate applications of Hermitian matrices in quantum mechanics
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Students and professionals in mathematics, particularly those studying linear algebra, as well as physicists and engineers working with quantum mechanics or systems involving Hermitian operators.

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Homework Statement


Prove that the eigenvalues of a Hermitian matrix is real.
http://www.proofwiki.org/wiki/Hermitian_Matrix_has_Real_Eigenvalues

The website says that "By Product with Conjugate Transpose Matrix is Hermitian, v*v is Hermitian. " where v* is the conjugate transpose of v.

Homework Equations





The Attempt at a Solution



I'm not sure why that is true. v*Av is equal to v*v, and v*Av is a Hermitian matrix. Intuitively, v*v seems like a Hermitian matrix, but I need a real theorem that would show that.
 
Physics news on Phys.org
Do you really need a proof that v*v is Hermitian? Some things are just obvious. This is one of those things.
 

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