The conjugate transpose of an matrix is formally defined by where the subscripts denote 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, bedaggered matrix, adjoint matrix or transjugate. The conjugate transpose of a matrix can be denoted by any of these symbols:
In some contexts, denotes the matrix with only complex conjugated entries and no transposition.
Example
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:
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.
Motivation
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 affected by complex z-multiplication on. An m-by-n matrix of complex numbers could therefore equally well be represented by a 2m-by-2n 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 n-by-m matrix made up of complex numbers.
The eigenvalues of are the complex conjugates of the eigenvalues of.
for any m-by-n matrix, any vector in and any vector. Here, denotes the standard complexinner product on, and similarly for.
Generalizations
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.
