Conjugate transpose
In mathematics, the conjugate transpose (or Hermitian transpose) of an mbyn matrix with complex entries, is the nbym matrix obtained from by taking the transpose and then taking the complex conjugate of each entry (the complex conjugate of being , for real numbers and ). It is often denoted as or .^{[1]}^{[2]}^{[3]}
For real matrices, the conjugate transpose is just the transpose, .
DefinitionEdit
The conjugate transpose of an matrix is formally defined by

(Eq.1) 
where the subscripts denote the th entry, for and , and the overbar denotes a scalar complex conjugate.
This definition can also be written as^{[3]}
where denotes the transpose and denotes the matrix with complex conjugated entries.
Other names for the conjugate transpose of a matrix are Hermitian conjugate, bedaggered matrix, adjoint matrix or transjugate. The conjugate transpose of a matrix can be denoted by any of these symbols:
 , commonly used in linear algebra^{[3]}
 , commonly used in linear algebra^{[1]}
 (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.
ExampleEdit
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 remarksEdit
A square matrix with entries is called
 Hermitian or selfadjoint 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 semidefinite 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.
MotivationEdit
The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2×2 real matrices, obeying matrix addition and multiplication:
That is, denoting each complex number z by the real 2×2 matrix of the linear transformation on the Argand diagram (viewed as the real vector space ), affected by complex zmultiplication on .
Thus, an mbyn matrix of complex numbers could be well represented by a 2mby2n 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 nbym matrix made up of complex numbers.
Properties of the conjugate transposeEdit
 for any two matrices and of the same dimensions.
 for any complex number and any mbyn matrix .
 for any mbyn matrix and any nbyp matrix . Note that the order of the factors is reversed.^{[2]}
 for any mbyn 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 mbyn matrix , any vector in and any vector . Here, denotes the standard complex inner product on , and similarly for .
GeneralizationsEdit
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 alsoEdit
ReferencesEdit
 ^ ^{a} ^{b} "Comprehensive List of Algebra Symbols". Math Vault. 20200325. Retrieved 20200908.
 ^ ^{a} ^{b} Weisstein, Eric W. "Conjugate Transpose". mathworld.wolfram.com. Retrieved 20200908.
 ^ ^{a} ^{b} ^{c} "conjugate transpose". planetmath.org. Retrieved 20200908.
External linksEdit
 "Adjoint matrix", Encyclopedia of Mathematics, EMS Press, 2001 [1994]