conjugate transpose

In mathematics, the conjugate transpose or adjoint 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^*) i,j = \overline{A j,i}

for 1 ≤ i ≤ n and 1 ≤ j ≤ m. This is a particular case of the Hermitian adjoint of a linear operator. More generally, if we have a linear map A from a complex vector space V to another W, the conjugate transpose of A is the conjugate of the transpose of A. It maps the conjugate dual of W to the conjugate dual of V. 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^* \qquad A^H \qquad A^\dagger\,.

Example

For example, if

A=\begin{bmatrix}3+i&2\\

2-2i&i\end{bmatrix} then

A^*=\begin{bmatrix}3-i&2+2i\\

2&-i\end{bmatrix}.

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. The square matrix A is called hermitian or self-adjoint if A = A^*. It is called normal if A^*A = AA^*. Even if A is not square, the two matrices A^*A and AA^* are both hermitian and in fact positive semi-definite. The adjoint matrix A^* should not be confused with the adjugate adj(A) (which in older texts is also sometimes called "adjoint").

Properties of the conjugate transpose

(A + B)^* = A^* + B^* for any two matrices A and B of the same format.
(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.
(A^*)^* = A for any matrix A.
If A is a square matrix, then det (A^*) = (det A)^*, trace (A^*) = (trace A)^*, and (A^*)^-1 = (A^-1)^*.
<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.

Adjoint operator in Hilbert space

The final property given above shows that if one views A as a linear operator from the Euclidean Hilbert space Cⁿ to C^m, then the matrix A^* corresponds to the adjoint operator. For an operator A on a Hilbert space H, the relation

\langle A x, y \rangle  = \langle x, A^* y \rangle

can be used to define the adjoint A^*, by means of the Riesz representation theorem. This definition can be extended even for operators which are not bounded. See self-adjoint operator for the details. The notation

A^{\dagger}

is also used to denote the adjoint of A, especially when used in conjunction with the bra-ket notation. The adjoint condition takes the form:

\lang \phi | (A|\psi) \rang =  \lang (A^{\dagger}\phi) | \psi) \rang.

The term Hermitian conjugate transpose is also sometimes used to refer to the adjoint. Although the etymology of this usage is not clear, it has been suggested that it results from the expression Hermitian operator being used to denote self-adjoint operators, that is operators A for which

A = A^{\dagger}

Note that there is a general theory of adjoint functors in category theory which includes the previous definition as a special case. See John Baez' expository article week78 for a discussion of this, and earlier writings for introductory material on category theory.

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