conjugate transpose

Definition If A is a complex matrix, then the conjugate transposeMathworldPlanetmath A is the matrix A=A¯T, where A¯ is the complex conjugateDlmfMathworldPlanetmath of A, and AT is the transposeMathworldPlanetmath of A.

It is clear that for real matrices, the conjugate transpose coincides with the transpose.

0.0.1 Properties

  1. 1.

    If A and B are complex matrices of same size, and α,β are complex constants, then

    (αA+βB) = α¯A+β¯B,
    A = A.
  2. 2.

    If A and B are complex matrices such that AB is defined, then

  3. 3.

    If A is a complex square matrixMathworldPlanetmath, then

    det(A) = detA¯,
    trace(A) = traceA¯,
    (A)-1 = (A-1),

    where trace and det are the trace and the determinantDlmfMathworldPlanetmath operators, and -1 is the inverse operator.

  4. 4.

    Suppose , is the standard inner product on n. Then for an arbitrary complex n×n matrix A, and vectors x,yn, we have



The conjugate transpose of A is also called the adjoint matrix of A, the Hermitian conjugate of A (whence one usually writes A=AH). The notation A is also used for the conjugate transpose [2]. In [1], A is also called the tranjugate of A.


  • 1 H. Eves, Elementary MatrixMathworldPlanetmath Theory, Dover publications, 1980.
  • 2 M. C. Pease, Methods of Matrix Algebra, Academic Press, 1965.

See also

  • Wikipedia, transpose

Title conjugate transpose
Canonical name ConjugateTranspose
Date of creation 2013-03-22 13:42:18
Last modified on 2013-03-22 13:42:18
Owner Koro (127)
Last modified by Koro (127)
Numerical id 10
Author Koro (127)
Entry type Definition
Classification msc 15-00
Classification msc 15A15
Synonym adjoint matrix
Synonym Hermitian conjugate
Synonym tranjugate
Related topic Transpose