Conjugate transpose

Encyclopedia : C : CO : CON : Conjugate transpose

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.

Contents

1 Example
2 Basic remarks
3 Motivation
4 Properties of the conjugate transpose
5 Generalizations
6 See also
7 External links

Example

[A=\begin3+i&2\\2-2i&i\end]

then

[A^*=\begin3-i&2+2i\\2&-i\end.]

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 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 is also sometimes called "adjoint").

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(\begin a & -b \\ b & a \end\Big) ]

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.

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. 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.

Generalizations

The last property given above shows that if one views A as a linear map 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.

External links

, [Conjugate Transpose] at MathWorld.
- [Conjugate transpose] on PlanetMath
From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.