Conjugate transpose

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

for 1 ≤ i ≤ n and 1 ≤ j ≤ m.

This definition can also be written as

<math> A^H \equiv {\overline A}^{T}</math>

where <math>A^T \,\!</math> denotes the transpose and <math> \overline A \,\!</math> 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:

<math>A^H \,\!</math>, commonly used in linear algebra
<math>A^\dagger \,\!</math>, universally used in quantum mechanics
<math>A^* \,\!</math> (which can also denote the complex conjugate, however)

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

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Example

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

then

<math>A^H=\begin{bmatrix}3-i&2+2i\\ 2&-i\end{bmatrix}.</math>

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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 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").

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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:

<math>a + ib \equiv \Big(\begin{matrix} a & -b \\ b & a \end{matrix}\Big) </math>

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 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.

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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.