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Mathematics for Physics

252 Hermitian Conjugate

A matrix obtained from another matrix by taking the transpose and then the complex conjugate of each entry that is used to define adjoints and to form norms of complex vectors.

definition [d] (Hermitian Conjugate = Adjoint = Conjugate Transpose = Hermitian Transpose) The matrix \(A^{\dagger}\) obtained from \(A\) by transposition together with complex conjugation:

  • (Matrix) \((A^{\dagger})_{ij} = \overline{a_{ji}}\) .
  • (Operator) \(\langle A\mathbf{a},\, \mathbf{b} \rangle = \langle \mathbf{a},\, A^{\dagger}\mathbf{b} \rangle\) for all \(\mathbf{a}, \mathbf{b}\).

where

  • \(A\) is a matrix on a complex Hilbert space.
  • \(A^{\dagger}\) is the Hermitian conjugate of \(A\).
  • \(a_{ij}\) are the entries of \(A\).
  • \(\overline{\,\cdot\,}\) denotes the complex conjugate.
  • \(\langle \cdot,\, \cdot \rangle\) is the inner product.
  • \(\mathbf{a}, \mathbf{b}\) are vectors in the Hilbert space.

Note:

  • the same definition applies when \(A\) is a linear operator.
  • \(A^{\dagger}\) is also written \(A^{*}\).
  • \(A^{\dagger}\) is also written \(A^{H}\).

252.1 Elementary Example

252.1.1 Simple

The Hermitian conjugate is the conjugate transpose of a matrix.

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

\[ A^{\dagger} = \begin{pmatrix} 1 & 0 \\ 2-i & 3 \end{pmatrix} \]

where

  • \(A^{\dagger}\) is the Hermitian conjugate of \(A\).
  • \((A^{\dagger})_{ij} = \overline{a_{ji}}\).

252.1.2 General

For a \(3 \times 3\) complex matrix, transpose and then conjugate every entry.

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

\[ A^{\dagger} = \begin{pmatrix} 1 & 2 & 0 \\ -i & 0 & 3 \\ 0 & 1-i & i \end{pmatrix} \]

where

  • \(A^{\dagger}\) is also written \(A^{*}\) or \(A^{H}\).

252.2 References

  1. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — matrix condition (2.53); operator adjoint (§5.3).
  2. Schwichtenberg, J. Physics from Symmetry. Springer, 2018. — dagger notation (\(U^{\dagger} = U^{T*}\), transpose plus complex conjugation).
  3. McWeeny, R., & Jones, H. Symmetry: An Introduction to Group Theory and Its Applications. 1963. — names “Hermitian transpose”, “adjoint”, “associate”.