Conjugate transpose

Summary

In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugation to each entry (the complex conjugate of being , for real numbers and ). There are several notations, such as or ,[1] ,[2] or (often in physics) .

For real matrices, the conjugate transpose is just the transpose, .

Definition edit

The conjugate transpose of an   matrix   is formally defined by

 

(Eq.1)

where the subscript   denotes the  -th entry, for   and  , and the overbar denotes a scalar complex conjugate.

This definition can also be written as

 

where   denotes the transpose and   denotes the matrix with complex conjugated entries.

Other names for the conjugate transpose of a matrix are Hermitian conjugate, adjoint matrix or transjugate. The conjugate transpose of a matrix   can be denoted by any of these symbols:

  •  , commonly used in linear algebra
  •  , commonly used in linear algebra
  •   (sometimes pronounced as A dagger), commonly used in quantum mechanics
  •  , although this symbol is more commonly used for the Moore–Penrose pseudoinverse

In some contexts,   denotes the matrix with only complex conjugated entries and no transposition.

Example edit

Suppose we want to calculate the conjugate transpose of the following matrix  .

 

We first transpose the matrix:

 

Then we conjugate every entry of the matrix:

 

Basic remarks edit

A square matrix   with entries   is called

  • Hermitian or self-adjoint if  ; i.e.,  .
  • Skew Hermitian or antihermitian if  ; i.e.,  .
  • Normal if  .
  • Unitary if  , equivalently  , equivalently  .

Even if   is not square, the two matrices   and   are both Hermitian and in fact positive semi-definite matrices.

The conjugate transpose "adjoint" matrix   should not be confused with the adjugate,  , which is also sometimes called adjoint.

The conjugate transpose of a matrix   with real entries reduces to the transpose of  , as the conjugate of a real number is the number itself.

The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by   real matrices, obeying matrix addition and multiplication:

 

That is, denoting each complex number   by the real   matrix of the linear transformation on the Argand diagram (viewed as the real vector space  ), affected by complex  -multiplication on  .

Thus, an   matrix of complex numbers could be well represented by a   matrix of real numbers. The conjugate transpose, therefore, arises very naturally as the result of simply transposing such a matrix—when viewed back again as an   matrix made up of complex numbers.


For an explanation of the notation used here, we begin by representing complex numbers   as the rotation matrix, that is,

 

Since   we are led to the matrix representations of the unit numbers as

  A general complex number   is then represented as

  The complex conjugate operation, where i→−i, is seen to be just the matrix transpose.

[3]

Properties of the conjugate transpose edit

  •   for any two matrices   and   of the same dimensions.
  •   for any complex number   and any   matrix  .
  •   for any   matrix   and any   matrix  . Note that the order of the factors is reversed.[1]
  •   for any   matrix  , i.e. Hermitian transposition is an involution.
  • If   is a square matrix, then   where   denotes the determinant of   .
  • If   is a square matrix, then   where   denotes the trace of  .
  •   is invertible if and only if   is invertible, and in that case  .
  • The eigenvalues of   are the complex conjugates of the eigenvalues of  .
  •   for any   matrix  , any vector in   and any vector  . Here,   denotes the standard complex inner product on  , and similarly for  .

Generalizations edit

The last property given above shows that if one views   as a linear transformation from Hilbert space   to   then the matrix   corresponds to the adjoint operator of  . The concept of adjoint operators between Hilbert spaces can thus be seen as a generalization of the conjugate transpose of matrices with respect to an orthonormal basis.

Another generalization is available: suppose   is a linear map from a complex vector space   to another,  , then the complex conjugate linear map as well as the transposed linear map are defined, and we may thus take the conjugate transpose of   to be the complex conjugate of the transpose of  . It maps the conjugate dual of   to the conjugate dual of  .

See also edit

References edit

  1. ^ a b Weisstein, Eric W. "Conjugate Transpose". mathworld.wolfram.com. Retrieved 2020-09-08.
  2. ^ H. W. Turnbull, A. C. Aitken, "An Introduction to the Theory of Canonical Matrices," 1932.
  3. ^ https://math.libretexts.org/Bookshelves/Differential_Equations/Applied_Linear_Algebra_and_Differential_Equations_(Chasnov)/02%3A_II._Linear_Algebra/01%3A_Matrices/1.06%3A_Matrix_Representation_of_Complex_Numbers

External links edit