Definitions

hermitian-conjugate

Conjugate transpose

In mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. The conjugate transpose is formally defined by
(A^*)_{ij} = overline{A_{ji}}
where the subscripts denote the i,j-th entry, for 1 ≤ in and 1 ≤ jm, and the overbar denotes a scalar complex conjugate. (The complex conjugate of a + bi, where a and b are reals, is a - bi.)

This definition can also be written as

A^* = (overline{A})^mathrm{T} = overline{A^mathrm{T}}
where A^mathrm{T} ,! denotes the transpose and overline A ,! denotes the matrix with complex conjugated entries.

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

Note that in some contexts A^* ,! can be used to denote the matrix with complex conjugated entries so care must be taken not to confuse notations.

Example

If
A = begin{bmatrix} 3 + i & 5 2-2i & i end{bmatrix}
then
A^* = begin{bmatrix} 3-i & 2+2i 5 & -i end{bmatrix}.

Basic remarks

If the entries of A are real, then A* coincides with the transpose AT 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 with entries a_{ij} is called

  • Hermitian or self-adjoint if A = A*, i.e., a_{ij}=a_{ji}^* ;
  • skew Hermitian or antihermitian if A = −A*, i.e., a_{ij}=-a_{ji}^{*} ;
  • normal if A*A = AA*.

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

The adjoint matrix A* 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 dimensions.
  • (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 tr(A*) = (tr A)*
  • A is invertible if and only if A* is invertible, and in that case we have (A*)−1 = (A−1)*.
  • The eigenvalues 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 Cn and any vector y in Cm. Here <·,·> denotes the standard complex inner product on Cm and Cn.

Generalizations

The last property given above shows that if one views A as a linear transformation from the Euclidean Hilbert space Cn to Cm, then the matrix A* 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

External links

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