Introduction: Matrices with Real Eigenvalues
A Hermitian matrix (self-adjoint matrix) equals its own adjoint: \(A = A^\dagger\). This seemingly simple property has profound consequences: Hermitian matrices always have real eigenvalues and orthogonal eigenvectors. In quantum mechanics, every measurable quantity corresponds to a Hermitian operator.
Definition and Conditions
A matrix \(A\) is Hermitian if \(A = A^\dagger\), meaning:
\[a_{ij} = \overline{a_{ji}}\]Consequences:
- Diagonal entries must be real (since \(a_{ii} = \overline{a_{ii}}\))
- Off-diagonal entries are complex conjugates of each other
The Spectral Theorem (for Hermitian Matrices)
If \(A\) is Hermitian:
- All eigenvalues are real
- Eigenvectors corresponding to different eigenvalues are orthogonal
- \(A\) can be diagonalized by a unitary matrix
Worked Examples
Example 1: Checking Hermiticity
\[A = \begin{pmatrix} 3 & 2-i \\ 2+i & 1 \end{pmatrix}\] \[A^\dagger = \begin{pmatrix} 3 & 2-i \\ 2+i & 1 \end{pmatrix} = A \checkmark\]This is Hermitian.
Example 2: Eigenvalues are Real
For the Pauli Z matrix:
\[\sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\]Eigenvalues: \(\lambda = 1\) and \(\lambda = -1\) (both real ✓)
Eigenvectors: \(|+\rangle = (1, 0)^T\) and \(|-\rangle = (0, 1)^T\) (orthogonal ✓)
Example 3: The Pauli X Matrix
\[\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \sigma_x^\dagger \checkmark\]Eigenvalues: \(\det(\sigma_x - \lambda I) = \lambda^2 - 1 = 0 \Rightarrow \lambda = \pm 1\)
The Quantum Connection
Every observable in quantum mechanics (position, momentum, energy, spin) is represented by a Hermitian operator. The requirement that eigenvalues be real ensures measurement outcomes are real numbers. The orthogonality of eigenstates means different outcomes are mutually exclusive—you can't get "position 5 and position 7" simultaneously.