Math Problem Statement
Define Hermitian matrix in Cn. Show that all the characteristic roots of a Hermitian matrix are real.
Solution
Definition of Hermitian Matrix in
A Hermitian matrix is a square matrix (complex -dimensional space) that satisfies the following condition:
where is the conjugate transpose (also known as Hermitian transpose) of . In other words, if , then:
where denotes the complex conjugate of . Hence, is Hermitian if:
Proof: Characteristic Roots of a Hermitian Matrix Are Real
Let be a Hermitian matrix, and let be a characteristic root (eigenvalue) of . This means there exists a non-zero vector such that:
To show that is real, consider the following steps:
-
Compute the inner product of both sides: Take the inner product of with :
Using the properties of inner products, .
Hence:
-
Express the inner product : Since is Hermitian, , and by the definition of the inner product:
Substituting , this becomes:
where is the complex conjugate of .
-
Equating the two expressions for : From step 1, , and from step 2, .
Therefore:
-
Simplify to conclude is real: Since is the norm of , it is a positive real number. Dividing through by :
This implies that is real.
Conclusion
Every characteristic root (eigenvalue) of a Hermitian matrix is real.
Let me know if you would like further clarifications or examples! Here are some related questions you might explore:
- What are some properties of eigenvalues and eigenvectors of Hermitian matrices?
- Can a Hermitian matrix have complex eigenvalues?
- How do Hermitian matrices relate to symmetric matrices over ?
- What is the spectral theorem for Hermitian matrices?
- How are unitary matrices connected to Hermitian matrices?
Tip: When working with Hermitian matrices, always leverage the fact that their eigenvectors corresponding to distinct eigenvalues are orthogonal.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Hermitian Matrices
Eigenvalues and Eigenvectors
Inner Product Spaces
Formulas
A = A* (Hermitian condition)
λ⟨v, v⟩ = ⟨Av, v⟩ (eigenvalue property)
Theorems
Hermitian Matrix Property: Eigenvalues of Hermitian matrices are real
Suitable Grade Level
Undergraduate
Related Recommendation
Hermitian Matrices and Real Eigenvalues in Complex Space
Hermitian Matrix with A² = I: Eigenvalues and Properties
Prove Real Eigenvalues of Hermitian Matrices and Nilpotency of T
Hermitian Matrix Eigenvalues, Trace and Rank Problems
Proof of Orthogonality for Eigenvectors of Symmetric Matrices with Distinct Eigenvalues