Definition of a Hermitian Matrix in $\mathbb{C}^n$:

A Hermitian matrix is a square matrix $A \in \mathbb{C}^{n \times n}$ (with complex entries) that satisfies the condition: $A = A^H,$ where $A^H$ denotes the conjugate transpose of $A$. That is, $A_{ij} = \overline{A_{ji}}$ for all $i$ and $j$, where $\overline{A_{ji}}$ is the complex conjugate of the element at position $(j, i)$.

Proof that all the characteristic roots (eigenvalues) of a Hermitian matrix are real:

Let $A$ be a Hermitian matrix, and let $\lambda$ be an eigenvalue of $A$, with the corresponding eigenvector $v \neq 0$. By definition of eigenvalues and eigenvectors: $A v = \lambda v.$

1. Take the conjugate transpose of the equation: $v^H A = \lambda^* v^H,$ where $v^H$ is the conjugate transpose of $v$, and $\lambda^*$ is the complex conjugate of $\lambda$.

2. Substitute $A = A^H$ (Hermitian property): $v^H A = v^H A^H = (A v)^H.$

   Substituting $A v = \lambda v$ into $(A v)^H$, we have: $v^H A = (\lambda v)^H = \lambda^* v^H.$

   Thus: $\lambda v^H v = \lambda^* v^H v,$ where $v^H v = \|v\|^2$ (the squared norm of $v$, which is a positive real number).

3. Simplify the equation: Since $\|v\|^2 > 0$, we divide by $\|v\|^2$: $\lambda = \lambda^*.$

4. Conclude that $\lambda$ is real: The equality $\lambda = \lambda^*$ implies that $\lambda$ is a real number.

Conclusion:

All eigenvalues (characteristic roots) of a Hermitian matrix are real.

Would you like additional clarification or detailed steps for any part? Let me know!

Here are 5 related questions for exploration:

1. How can you verify whether a given matrix is Hermitian?
2. What is the relationship between Hermitian matrices and positive semidefinite matrices?
3. What are some applications of Hermitian matrices in quantum mechanics or physics?
4. How do Hermitian matrices relate to symmetric matrices over real numbers?
5. Can Hermitian matrices have complex eigenvectors, even though their eigenvalues are real?

Tip: When solving problems involving Hermitian matrices, always remember that the eigenvectors corresponding to distinct eigenvalues are orthogonal!