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

<math>(A^*)[i,j] = \overline{A[j,i]}<math>

for 1 ≤ i ≤ n and 1 ≤ j ≤ m. This is a particular case of the Hermitian conjugate (sometimes called Hermitian adjoint or just adjoint) linear operator.

Example

For example, if

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

2-2i&i\end{bmatrix}<math> then

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

2&-i\end{bmatrix}<math>

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.
• <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.

In fact it can be used to define what is meant by that. Assuming now we are in a Hilbert space H, the relation

<Ax,y> = <x, A^*y>

can be used to define the adjoint operator A^*, by means of the Riesz representation theorem.

When working in Hilbert space, especially with the bra-ket notation, the adjoint operator - called the Hermitian Conjugate, denoted as <math>A^{\dagger}<math>, is defined by the relation

<math> \lang \phi | (A|\psi) \rang = \lang (A^{\dagger}\phi) | \psi) \rang <math>

The term Hermitian conjugate transpose is used since if <math> A = A^{\dagger} <math> than A is called an Hermitian operator.

It has also been stated in which way the above can be related to the notion of a pair of adjoint functors in category theory. An explanation is given by John Baez in of his famous series. See also the previous weeks for a gentle introduction to category theory.