yawiki.org entry for Conjugate transpose

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In mathematics, the conjugate transpose or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A^{obtained from A by taking the transpose and then taking the complex conjugate of each entry. Formally: $(A^$ )_ = overline

where the subscripts denote the i,j-th entry, for 1 ≤ i ≤ n and 1 ≤ j ≤ m.

This definition can also be written as

$A^$ equiv ^

where $A^T ,!$ denotes the transpose and $overline A ,!$ denotes the matrix with complex conjugated entries.

Alternative names for the conjugate transpose of a matrix are adjoint matrix, Hermitian conjugate, or tranjugate. The conjugate transpose of a matrix A can be denoted by any of these symbols:

$A^$ ,! or $A^H ,!$ , commonly used in linear algebra

$A^dagger ,!$ , universally used in quantum mechanics

Note that in some contexts $A^$ ,! can be used to denote the complex conjugate so care must be taken not to confuse notations.

    Conjugate transpose

        Example

        Basic remarks

        Motivation

        Properties of the conjugate transpose

        Generalizations

        See also

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Example

If

$A=egin3+i&5\
2-2i&iend$

then

$A^$ =egin3-i&2+2i\
5&-iend.

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Basic remarks

If the entries of A are real, then A^{coincides with the transpose A^T of A. It is often useful to think of square complex matrices as "generalized complex numbers", and of the conjugate transpose as a generalization of complex conjugation.

A square matrix A is called

Hermitian or self-adjoint if A = A^;
skew Hermitian if A = −A^;
normal if AA = AA.

Even if A is not square, the two matrices AA and AA are both Hermitian and in fact positive semi-definite matrices.

The adjoint matrix A^{should not be confused with the adjugate adj(A) (which is also sometimes called "adjoint").

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Motivation

The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2×2 skew-symmetric matrices, obeying matrix addition and multiplication:

$a + ib equiv Big(egin a & -b \ b & a endBig)$

An m-by-n matrix of complex numbers could therefore equally well be represented by a 2m-by-2n matrix of real numbers. It therefore arises very naturally that when transposing such a matrix which is made up of complex numbers, one may in the process also have to take the complex conjugate of each entry.

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Properties of the conjugate transpose

(A + B)^{= A^{+ B^{for any two matrices A and B of the same dimensions.}}}
(rA)^{= r^{A^{for any complex number r and any matrix A. Here r^{refers to the complex conjugate of r.}}}}
(AB)^{= B^{A^{for any m-by-n matrix A and any n-by-p matrix B. Note that the order of the factors is reversed.}}}
(A^{)^{= A for any matrix A.}}
If A is a square matrix, then det (A^{) = (det A)^{and trace (A^{) = (trace A)}}}
A is invertible if and only if A^{is invertible, and in that case we have (A^{)^-1 = (A^-1)^.}}
The eigenvalues of A^{are the complex conjugates of the eigenvalues of A.}
<Ax,y> = <x, A^{y> for any m-by-n matrix A, any vector x in Cⁿ and any vector y in C^m. Here <.,.> denotes the ordinary Euclidean inner product (or dot product) on C^m and Cⁿ.}

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Generalizations

The last property given above shows that if one views A as a linear transformation from the Euclidean Hilbert space Cⁿ to C^m, then the matrix A^{corresponds to the adjoint operator of A. The concept of adjoint operators between Hilbert spaces can thus be seen as a generalization of the conjugate transpose of matrices.

Another generalization is available: suppose A is a linear map from a complex vector space V to another W, then the complex conjugate linear map as well as the transposed linear map are defined, and we may thus take the conjugate transpose of A to be the complex conjugate of the transpose of A. It maps the conjugate dual of W to the conjugate dual of V.

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See also

Hermitian conjugate}}}}

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