Conjugate transpose
In mathematics, the conjugate transpose (or Hermitian transpose) of an "m"-by-"n" matrix formula_1 with complex entries is the "n"-by-"m" matrix obtained from formula_1 by taking the transpose and then taking the complex conjugate of each entry (the complex conjugate of formula_3 being formula_4, for real numbers formula_5 and formula_6). It is often denoted as formula_7 or formula_8. For real matrices, the conjugate transpose is just the transpose, formula_9. Definition. The conjugate transpose of an formula_10 matrix formula_1 is formally defined by where the subscript formula_12 denotes the formula_13-th entry, for formula_14 and formula_15, and the overbar denotes  a scalar complex conjugate. This definition can also be written as where formula_17 denotes the transpose and formula_18 denotes the matrix with complex conjugated entries. Other names for the conjugate transpose of a matrix are Hermitian conjugate, adjoint matrix or transjugate. The conjugate transpose of a matrix formula_1 can be denoted by any of these symbols: In some contexts, formula_8 denotes the matrix with only complex conjugated entries and no transposition. Example. Suppose we want to calculate the conjugate transpose of the following matrix formula_1. We first transpose the matrix: Then we conjugate every entry of the matrix: Basic remarks . A square matrix formula_1 with entries formula_30 is called Even if formula_1 is not square, the two matrices formula_40 and formula_41 are both Hermitian and in fact positive semi-definite matrices. The conjugate transpose "adjoint" matrix formula_7 should not be confused with the adjugate, formula_43, which is also sometimes called "adjoint". The conjugate transpose of a matrix formula_1 with real entries reduces to the transpose of formula_1, 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 (viewed as the "real" vector space formula_47), affected by complex "z"-multiplication on formula_48. Thus, an "m"-by-"n" matrix of complex numbers could be well represented by a 2"m"-by-2"n" 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. Generalizations. The last property given above shows that if one views formula_1 as a linear transformation from  Hilbert space formula_81 to formula_82 then the matrix formula_7 corresponds to the adjoint operator of formula_84. 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 formula_85 is a linear map from a complex vector space formula_86 to another, formula_87, then the complex conjugate linear map as well as the transposed linear map are defined, and we may thus take the conjugate transpose of formula_85 to be the complex conjugate of the transpose of formula_85. It maps the  conjugate dual of formula_87 to the conjugate dual of formula_86.