Recall that a matrix is Hermitian (or self-adjoint) if $A=\adjoint{A}$ (Definition HM). In the case where $A$ is a matrix whose entries are all real numbers, being Hermitian is identical to being symmetric (Definition SYM). Keep this in mind as you read the next two theorems. Their hypotheses could be changed to “suppose $A$ is a real symmetric matrix.”
Suppose that $A$ is a Hermitian matrix and $\lambda$ is an eigenvalue of $A$. Then $\lambda\in{\mathbb R}$.
Notice the appealing symmetry to the justifications given for the steps of this proof. In the center is the ability to pitch a Hermitian matrix from one side of the inner product to the other.
Look back and compare Example ESMS4 and Example CEMS6. In Example CEMS6 the matrix has only real entries, yet the characteristic polynomial has roots that are complex numbers, and so the matrix has complex eigenvalues. However, in Example ESMS4, the matrix has only real entries, but is also symmetric, and hence Hermitian. So by Theorem HMRE, we were guaranteed eigenvalues that are real numbers.
In many physical problems, a matrix of interest will be real and symmetric, or Hermitian. Then if the eigenvalues are to represent physical quantities of interest, Theorem HMRE guarantees that these values will not be complex numbers.
The eigenvectors of a Hermitian matrix also enjoy a pleasing property that we will exploit later.
Suppose that $A$ is a Hermitian matrix and $\vect{x}$ and $\vect{y}$ are two eigenvectors of $A$ for different eigenvalues. Then $\vect{x}$ and $\vect{y}$ are orthogonal vectors.
Notice again how the key step in this proof is the fundamental property of a Hermitian matrix (Theorem HMIP) — the ability to swap $A$ across the two arguments of the inner product. We will build on these results and continue to see some more interesting properties in Section OD.