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Mathematics for Physics

244 Conjugate Transpose

A matrix obtained by interchanging rows and columns and then taking the complex conjugate of each entry that is used to form adjoints of complex matrices.

Note: Also called Hermitian conjugate. Also called Hermitian transpose. Also called adjoint of a matrix.

definition [d] (Conjugate Transpose) From Axler: the conjugate transpose of an \(m\)-by-\(n\) matrix \(A\) is the \(n\)-by-\(m\) matrix \(A^{*}\) obtained by interchanging the rows and columns and then taking the complex conjugate of each entry. In other words, if \(j \in \{1,\ldots,n\}\) and \(k \in \{1,\ldots,m\}\), then

  • \((A^{*})_{j,k} = \overline{A_{k,j}}\) .

where

  • \(A\) is an \(m\)-by-\(n\) matrix with complex entries.
  • \(A^{*}\) is the conjugate transpose of \(A\).
  • \(\overline{\,\cdot\,}\) denotes the complex conjugate.

definition [d] (Hermitian Transpose) From Cohen: the Hermitian transpose, often called simply the Hermitian, is a conjugate-and-transpose operation. It is indicated with a superscripted \(H\) instead of a \(T\).

where

  • \(A^{H}\) denotes the Hermitian transpose of a matrix \(A\).
  • \(v^{H}\) denotes the Hermitian transpose of a vector \(v\).

definition [d] (Adjoint of a Matrix) From Arfken: the adjoint of a matrix \(A\), denoted \(A^{\dagger}\), is obtained by both complex conjugating and transposing it. Thus

  • \((A^{\dagger})_{ij} = a_{ji}^{*}\) .

where

  • \(A\) is a matrix.
  • \(A^{\dagger}\) is the adjoint of \(A\) in the matrix sense.
  • \(a_{ji}^{*}\) is the complex conjugate of the entry \(a_{ji}\).

244.1 Elementary Example

244.1.1 Simple

Interchange rows and columns, then conjugate each entry.

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

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

where

  • \(A^{*}\) is the conjugate transpose of \(A\).
  • \((A^{*})_{j,k} = \overline{A_{k,j}}\).

244.1.2 General

For a \(3 \times 2\) matrix, the conjugate transpose is \(2 \times 3\).

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

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

where

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

244.2 References

  1. Axler, S. Linear Algebra Done Right. Springer, 2024. — Definition 7.7: conjugate transpose \(A^{*}\), \((A^{*})_{j,k} = \overline{A_{k,j}}\).
  2. Cohen, M. X. Linear Algebra: Theory, Intuition, Code. Sincxpress BV, 2021. — Hermitian transpose \(A^{H}\).
  3. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — \((A^{\dagger})_{ij} = a_{ji}^{*}\).