English

Noun

transpose conjugate ‎(plural transpose conjugates)

1. (mathematics) The transpose of a matrix, after replacing each element with its complex conjugate.
   • 2006, David Andrews, Lumped Element Quadrature Hybrids, page 22,

     Further constraints upon the four-port are discovered when the elements equating to zero in the product of the unitary S-matrix with its transpose conjugate are considered.

   • 2012, Steven B. Damelin, Willard Miller, Jr., The Mathematics of Signal Processing, page 299,

     In that case the inverse of the matrix is just the transpose conjugate and solving for the original signal x from the truncated outputs {y⁽⁰⁾(2n)},{y⁽¹⁾(2n)} is simple.

   • 2012, Dennis M. Sullivan, Quantum Mechanics for Electrical Engineers, page 85,

     Throughout this text, the transpose conjugate of a matrix or vector is given by A⁺ or b⁺, respectively.

Usage notes

In linear algebra, it may be denoted using the notations A^*, A^H, A^† or A⁺. Of these, the superscripted dagger (glyph) notation is universally used in quantum mechanics. Also, the notation A⁺ is commonly used for the Moore–Penrose pseudoinverse.

In some contexts, A^* denotes the matrix of complex conjugate entries, and the transpose conjugate is then denoted by A^*T or A^T*.

Synonyms

• (transpose of matrix of complex conjugates): adjoint matrix, bedaggered matrix, conjugate transpose, Hermitian conjugate, Hermitian transpose, transjugate

See also

• adjugate matrix (historically also sometimes called "adjoint")