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 being , for real numbers and ). It is often denoted as or .
For real matrices, the conjugate transpose is just the transpose, .
The conjugate transpose of an matrix is formally defined by
where the subscript denotes the -th entry, for and , and the overbar denotes a scalar complex conjugate.
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.
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 .
Thus, an m-by-n matrix of complex numbers could be well 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 an n-by-m 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 m-by-n matrix .
for any m-by-n matrix and any n-by-p matrix . Note that the order of the factors is reversed.
for any m-by-n 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 .
for any m-by-n matrix , any vector in and any vector . Here, denotes the standard complex inner product on , and similarly for .
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 .