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, bedaggered matrix, 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 "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.