Proving that the eigenvalues of a Hermitian matrix is real

In summary, the website provides a proof that the eigenvalues of a Hermitian matrix are real. This is based on the fact that the product of a Hermitian matrix with its conjugate transpose is also a Hermitian matrix. The conversation also discusses the intuition behind this concept and questions the need for a formal proof.
  • #1
stgermaine
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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.
 
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  • #2
Do you really need a proof that v*v is Hermitian? Some things are just obvious. This is one of those things.
 

1. What is a Hermitian matrix?

A Hermitian matrix is a square matrix with complex entries that is equal to its own conjugate transpose. This means that the matrix is symmetric along the main diagonal and has conjugate pairs of entries above and below the diagonal.

2. Why is it important to prove that the eigenvalues of a Hermitian matrix are real?

Proving that the eigenvalues of a Hermitian matrix are real is important because it guarantees that the matrix is diagonalizable and that its eigenvectors form an orthogonal basis. This allows us to simplify calculations and make further deductions about the matrix.

3. How can we prove that the eigenvalues of a Hermitian matrix are real?

We can prove that the eigenvalues of a Hermitian matrix are real by using the spectral theorem, which states that for any Hermitian matrix, its eigenvalues are all real and its eigenvectors form an orthogonal basis. We can also use the fact that the determinant of a Hermitian matrix is always real.

4. What does it mean for the eigenvalues of a Hermitian matrix to be real?

When the eigenvalues of a Hermitian matrix are real, it means that they do not have any imaginary components. This also implies that the matrix is symmetric and has real entries along the main diagonal.

5. Can a non-Hermitian matrix have real eigenvalues?

No, a non-Hermitian matrix cannot have real eigenvalues. This is because the eigenvalues of a non-Hermitian matrix may have imaginary components, which violates the definition of a Hermitian matrix. However, a real symmetric matrix is a special case of a Hermitian matrix and can have real eigenvalues.

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