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:(A^H) = \overline{A[j,i }
for 1 ≤ ''i'' ≤ ''n'' and 1 ≤ ''j'' ≤ ''m''.

This definition can also be written as
: A^H \equiv {\overline A}^{T}
where A^T \,\! denotes the transpose and \overline A \,\! denotes the matrix with complex conjugated entries.

Alternative names for the conjugate transpose of a matrix are adjoint matrix, '''Hermitian conjugate''', or '''tranjugate'''. The conjugate transpose of a matrix ''A'' can be denoted by any of these symbols:
  • A^H \,\!, commonly used in Linear Algebra

  • A^\dagger \,\!, universally used in Quantum Mechanics

  • A^--- \,\! (which can also denote the complex conjugate, however)



EXAMPLE


If

A=\begin{bmatrix}3+i&2\
2-2i&i\end{bmatrix}

then

A^H=\begin{bmatrix}3-i&2+2i\
2&-i\end{bmatrix}.


BASIC REMARKS


  • coincides with the transpose ''A''T of ''A''. It is often useful to think of Square complex matrices as "generalized complex numbers", and of the conjugate transpose as a generalization of Complex Conjugation .


A square matrix ''A'' is called


  • should not be confused with the Adjugate adj(''A'') (which is also sometimes called "adjoint").



MOTIVATION


The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2×2 skew-symmetric matrices, obeying matrix addition and multiplication:

:a + ib \equiv \Big(\begin{matrix} a & -b \ b & a \end{matrix}\Big)

An ''m''-by-''n'' matrix of complex numbers could therefore equally well be represented by a ''2m''-by-''2n'' matrix of real numbers. It therefore arises very naturally that when transposing such a matrix which is made up of complex numbers, one may in the process also have to take the complex conjugate of each entry.


PROPERTIES OF THE CONJUGATE TRANSPOSE


  • (''A'' + ''B'')--- = ''A''--- + ''B''--- for any two matrices ''A'' and ''B'' of the same format.

  • (''rA'')--- = ''r''---''A''--- for any complex number ''r'' and any matrix ''A''. Here ''r''--- refers to the complex conjugate of ''r''.

  • (''AB'')--- = ''B''---''A''--- for any ''m''-by-''n'' matrix ''A'' and any ''n''-by-''p'' matrix ''B''. Note that the order of the factors is reversed.

  • (''A''---)--- = ''A'' for any matrix ''A''.

  • If ''A'' is a square matrix, then Det (''A''---) = (det A)--- and Trace (''A''---) = (trace A)---

  • ''A'' is Invertible if and only if ''A''--- is invertible, and in that case we have (''A''---)-1 = (''A''-1)---.

  • The Eigenvalue s of ''A''--- are the complex conjugates of the eigenvalues of ''A''.

  • <''Ax'',''y''> = <''x'', ''A''---''y''> for any ''m''-by-''n'' matrix ''A'', any vector ''x'' in C''n'' and any vector ''y'' in C''m''. Here <.,.> denotes the ordinary Euclidean Inner Product (or dot product) on C''m'' and C''n''.



GENERALIZATIONS


  • corresponds to the Adjoint Operator of ''A''. The concept of adjoint operators between Hilbert spaces can thus be seen as a generalization of the conjugate transpose of matrices.


Another generalization is available: suppose ''A'' is a linear map from a complex Vector Space ''V'' to another ''W'', then the Complex Conjugate Linear Map as well as the Transposed Linear Map are defined, and we may thus take the conjugate transpose of ''A'' to be the complex conjugate of the transpose of ''A''. It maps the conjugate Dual of ''W'' to the conjugate dual of ''V''.


SEE ALSO



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