"Adjoint matrix" redirects here. For the transpose of cofactor, see Adjugate matrix.
In mathematics, the conjugate transpose or Hermitian transpose of an m-by-nmatrix with complex entries is the n-by-m matrix obtained from by taking the transpose and then taking the complex conjugate of each entry. (The complex conjugate of , where and are real numbers, is .)
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:
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 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 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 .
This page is based on a Wikipedia article written by contributors (read/edit). Text is available under the CC BY-SA 4.0 license; additional terms may apply. Images, videos and audio are available under their respective licenses.