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Mathematics for Physics

251 Hermitian

A matrix equal to its conjugate transpose that is used to guarantee real eigenvalues.

Note: Also called self-adjoint. Also called Hermitian operator when stated for a linear transformation on an inner product space.

definition [d] (Hermitian = Hermitian Matrix) From Arfken: a matrix \(H\) must be square, and its elements must satisfy

  • \((H^{\dagger})_{ij} = (H)_{ij}\) ,

which means

  • \(h_{ji}^{*} = h_{ij}\) .

where

  • \(H\) is a square matrix.
  • \(H^{\dagger}\) is the Hermitian conjugate of \(H\).
  • \(h_{ij}\) are the entries of \(H\).
  • \(h_{ji}^{*}\) is the complex conjugate of the entry \(h_{ji}\).

definition [d] (Hermitian = Self-Adjoint Operator) From Kreyszig: a bounded linear operator \(T: H \rightarrow H\) on a Hilbert space \(H\) is said to be self-adjoint, written also Hermitian, if

  • \(T^{*} = T\) .

where

  • \(H\) is a Hilbert space.
  • \(T\) is a bounded linear operator on \(H\).
  • \(T^{*}\) is the adjoint of \(T\).

Note:

  • Hall: what a mathematician calls the adjoint and writes \(A^{*}\), a physicist calls the Hermitian conjugate and writes \(A^{\dagger}\); physicists call self-adjoint operators Hermitian.

251.1 Elementary Example

251.1.1 Simple

A \(2 \times 2\) matrix is Hermitian when it equals its conjugate transpose.

\[ H = \begin{pmatrix} 1 & 1+i \\ 1-i & 2 \end{pmatrix} \]

\[ H^{\dagger} = \begin{pmatrix} 1 & 1+i \\ 1-i & 2 \end{pmatrix} = H \]

where

  • \(H\) is the Hermitian matrix.
  • \(H^{\dagger}\) is the Hermitian conjugate of \(H\).
  • \(i\) is the imaginary unit with \(i^{2} = -1\).

251.1.2 General

A \(3 \times 3\) Hermitian matrix has real diagonal entries, and off-diagonal pairs are conjugates of each other.

\[ H = \begin{pmatrix} 2 & 1-i & 0 \\ 1+i & 3 & 4i \\ 0 & -4i & 1 \end{pmatrix} \]

\[ H^{\dagger} = H \]

\[ h_{ji}^{*} = h_{ij} \]

where

  • \(h_{ij}\) are the entries of \(H\).
  • \(h_{11}, h_{22}, h_{33}\) are real.
  • \(T^{*} = T\) is the same condition for the operator represented by \(H\).

251.2 References

  1. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — Hermitian matrix: \((H^{\dagger})_{ij} = (H)_{ij}\), equivalently \(h_{ji}^{*} = h_{ij}\).
  2. Kreyszig, E. Introductory Functional Analysis with Applications. Wiley, 1989. — self-adjoint or Hermitian operator: \(T^{*} = T\).
  3. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — adjoint \(A^{*}\) versus Hermitian conjugate \(A^{\dagger}\); self-adjoint called Hermitian in physics.