Кешенді матрицаның түрлендірілген қосымшасы

Complex matrix A* obtained from a matrix A by transposing it and conjugating each entry

In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugation to each entry (the complex conjugate of being , for real numbers and ). There are several notations, such as or , , or (often in physics)
For real matrices, the conjugate transpose is just the transpose, .

for any two matrices and of the same dimensions. for any complex number and any matrix for any matrix and any matrix Note that the order of the factors is reversed. for any 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 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 .