Conjugate transpose

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In mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, bedaggered matrix, 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 (i.e., negating their imaginary parts but not their real parts). The conjugate transpose is formally defined by

(\mathbf{A}^*)_{ij} = \overline{\mathbf{A}_{ji}}

where the subscripts denote the i,j-th entry, for 1 ≤ in and 1 ≤ jm, and the overbar denotes a scalar complex conjugate. (The complex conjugate of a + bi, where a and b are reals, is a - bi.)

This definition can also be written as

\mathbf{A}^* = (\overline{\mathbf{A}})^\mathrm{T} = \overline{\mathbf{A}^\mathrm{T}}

where \mathbf{A}^\mathrm{T} \,\! denotes the transpose and \overline{\mathbf{A}} \,\! denotes the matrix with complex conjugated entries.

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

In some contexts, \mathbf{A}^* \,\! denotes the matrix with complex conjugated entries, and the conjugate transpose is then denoted by \mathbf{A}^{*\mathrm{T}} \,\! or \mathbf{A}^{\mathrm{T}*} \,\!.

Example[edit]

If

\mathbf{A} = \begin{bmatrix} 1 & -2-i \\ 1+i & i \end{bmatrix}

then

\mathbf{A}^* = \begin{bmatrix} 1 & 1-i \\ -2+i & -i\end{bmatrix}

Basic remarks[edit]

A square matrix A with entries a_{ij} is called

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

The conjugate transpose "adjoint" matrix A* should not be confused with the adjugate adj(A), which is also sometimes called "adjoint".

Finding the conjugate transpose of a matrix A with real entries reduces to finding the transpose of A, as the conjugate of a real number is the number itself.

Motivation[edit]

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

a + ib \equiv  \left(\begin{matrix} a & -b \\ b & a \end{matrix}\right).

That is, denoting each complex number z by the real 2×2 matrix of the linear transformation on the Argand diagram (viewed as the real vector space \mathbb{R}^2) affected by complex z-multiplication on \mathbb{C}.

An m-by-n matrix of complex numbers could therefore equally well be represented by a 2m-by-2n matrix of real numbers. The conjugate transpose therefore arises very naturally as the result of simply transposing such a matrix, when viewed back again as n-by-m matrix made up of complex numbers.

Properties of the conjugate transpose[edit]

  • (A + B)* = A* + B* for any two matrices A and B of the same dimensions.
  • (r A)* = 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 tr(A*) = (tr A)*
  • A is invertible if and only if A* is invertible, and in that case (A*)−1 = (A−1)*.
  • The eigenvalues of A* are the complex conjugates of the eigenvalues of A.
  • \langle \mathbf{Ax}, \mathbf{y}\rangle = \langle \mathbf{x},\mathbf{A}^* \mathbf{y} \rangle for any m-by-n matrix A, any vector x in  \mathbb{C}^n and any vector y in  \mathbb{C}^m . Here, \langle\cdot,\cdot\rangle denotes the standard complex inner product on  \mathbb{C}^m and  \mathbb{C}^n .

Generalizations[edit]

The last property given above shows that if one views A as a linear transformation from Euclidean Hilbert space  \mathbb{C}^n to  \mathbb{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.

See also[edit]

External links[edit]