Conjugate transpose explained

In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an

m x n complex matrix A is an n x m matrix obtained by transposing A and applying complex conjugation to each entry (the complex conjugate of a + ib being a - ib , for real numbers a and b ). There are several notations, such as A ^{H} or A ^{*} ,^[1] A ' ,^[2] or (often in physics) A ^{\dagger} .

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

A ^{H} = A ^{\operatorname{T}} .

Definition

The conjugate transpose of an

m x n matrix A is formally defined by

where the subscript

ij denotes the (i,j) -th entry, for 1 \le i \le n and 1 \le j \le m , and the overbar denotes a scalar complex conjugate.

This definition can also be written as

A ^{H} = \left(\overline{ A }\right)^\operatorname = \overline

where

A ^{\operatorname{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, adjoint matrix or transjugate. The conjugate transpose of a matrix

A can be denoted by any of these symbols:A ^{*} , commonly used in linear algebraA ^{H} , commonly used in linear algebraA ^{\dagger} (sometimes pronounced as A dagger), commonly used in quantum mechanicsA ^{+} , although this symbol is more commonly used for the Moore–Penrose pseudoinverse

In some contexts,

A ^{*} denotes the matrix with only complex conjugated entries and no transposition.

Example

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

A .A = \begin{bmatrix} 1 & -2 - i & 5 \ 1 + i & i & 4-2 i \end{bmatrix} We first transpose the matrix:A ^{\operatorname{T}} = \begin{bmatrix} 1 & 1 + i \ -2 - i & i \ 5 & 4-2 i\end{bmatrix} Then we conjugate every entry of the matrix:A ^{H} = \begin{bmatrix} 1 & 1 - i \ -2 + i & - i \ 5 & 4+2 i\end{bmatrix}

Basic remarks

A square matrix

A with entries a _{ij} is called

• Hermitian or self-adjoint if

A = A ^{H} ; i.e., a _{ij} = \overline{a _{ji} }.

• Skew Hermitian or antihermitian if

A =- A ^{H} ; i.e., a _{ij} = - \overline{a _{ji} }.

• Normal if

A ^{H} A = A A ^{H} .

• Unitary if

A ^{H} = A ^{-1} , equivalently A A ^{H} = \boldsymbol{I} , equivalently A ^{HA} = \boldsymbol{I} .

Even if

A is not square, the two matrices A ^{HA} and A A ^{H} are both Hermitian and in fact positive semi-definite matrices.

The conjugate transpose "adjoint" matrix

A ^{H} should not be confused with the adjugate, \operatorname{adj}( A ) , which is also sometimes called adjoint.

The conjugate transpose of a matrix

A with real entries reduces to the transpose of A , 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

2 x 2 real matrices, obeying matrix addition and multiplication:a + ib \equiv \begin{bmatrix} a & - b \ b & a \end{bmatrix}.

That is, denoting each complex number

z by the real 2 x 2 matrix of the linear transformation on the Argand diagram (viewed as the real vector space R ^{2} ), affected by complex z -multiplication on C .

Thus, an

m x n matrix of complex numbers could be well represented by a 2 m x 2 n 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 n x m matrix made up of complex numbers.

For an explanation of the notation used here, we begin by representing complex numbers

e ^{i\theta} as the rotation matrix, that is,e ^{i\theta} = \begin{pmatrix} \cos \theta & - \sin \theta \ \sin \theta & \cos \theta \end{pmatrix} = \cos \theta \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} + \sin \theta \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix}.

Since

e ^{i\theta} = \cos \theta + i \sin \theta we are led to the matrix representations of the unit numbers as1 = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}, i = \begin{pmatrix} 0 & -1 \ 1 & 0 \end{pmatrix}. A general complex number z = x + iy is then represented asz = \begin{pmatrix} x & - y \ y & x \end{pmatrix}. The complex conjugate operation, where i→−i, is seen to be just the matrix transpose.

^[3]

Properties of the conjugate transpose

( A + \boldsymbol{B}) ^{H} = A ^{H} + \boldsymbol{B} ^{H} for any two matrices A and \boldsymbol{B} of the same dimensions.(z A ) ^{H} = \overline{z} A ^{H} for any complex number z and any m x n matrix A .( A \boldsymbol{B}) ^{H} = \boldsymbol{B} ^{H} A ^{H} for any m x n matrix A and any n x p matrix \boldsymbol{B} . Note that the order of the factors is reversed.\left( A ^{H\right)} ^{H} = A for any m x n matrix A , i.e. Hermitian transposition is an involution.

• If

A is a square matrix, then \det\left( A ^{H\right)} = \overline{\det\left( A \right)} where \operatorname{det}(A) denotes the determinant of A .

• If

A is a square matrix, then \operatorname{tr}\left( A ^{H\right)} = \overline{\operatorname{tr}( A )} where \operatorname{tr}(A) denotes the trace of A .A is invertible if and only if A ^{H} is invertible, and in that case \left( A ^{H\right)} ^{-1} = \left( A ^{-1} \right) ^{H} .

• The eigenvalues of

A ^{H} are the complex conjugates of the eigenvalues of A .\left\langle A x,y \right\rangle _{m} = \left\langle x, A ^{H} y\right\rangle _{n} for any m x n matrix A , any vector in x \in C ^{n} and any vector y \in C ^{m} . Here, \langle ⋅ , ⋅ \rangle _{m} denotes the standard complex inner product on C ^{m} , and similarly for \langle ⋅ , ⋅ \rangle _{n} .

Generalizations

The last property given above shows that if one views

A as a linear transformation from Hilbert space C ^{n} to C ^{m} , then the matrix A ^{H} 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 with respect to an orthonormal basis.

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

• Complex dot product
• Hermitian adjoint
• Adjugate matrix

References

1. Web site: Weisstein. Eric W.. Conjugate Transpose. 2020-09-08. mathworld.wolfram.com. en.
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