Conjugate transpose

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In mathematics, the conjugate transpose or Hermitian transpose 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. (The complex conjugate of a + bi, where a and b are reals, is abi.) The conjugate transpose is formally defined by

where the subscripts denote the (i,j)-th entry, for 1 ≤ in and 1 ≤ jm, and the overbar denotes a scalar complex conjugate.

This definition can also be written as

where denotes the transpose and denotes the matrix with complex conjugated entries.

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

  • or , commonly used in linear algebra
  • (sometimes pronounced as "A dagger"), universally used in quantum mechanics
  • , although this symbol is more commonly used for the Moore–Penrose pseudoinverse

In some contexts, denotes the matrix with complex conjugated entries, and the conjugate transpose is then denoted by or .

Example[edit]

If

then

Basic remarks[edit]

A square matrix A with entries is called

  • Hermitian or self-adjoint if A = A, i.e.  .
  • skew Hermitian or antihermitian if A = −A, i.e.  .
  • normal if AA = AA.
  • unitary if A = A−1.

Even if A is not square, the two matrices AA 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".

The conjugate transpose of a matrix A with real entries reduces to 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:

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 ) affected by complex z-multiplication on .

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.
  • (rA) = rA for any complex number r and any m-by-n matrix A.
  • (AB) = BA 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 m-by-n 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.
  • Ax, y⟩ = ⟨x, Ay for any m-by-n matrix A, any vector x in and any vector y in . Here, denotes the standard complex inner product on and .

Generalizations[edit]

The last property given above shows that if one views A as a linear transformation from Euclidean Hilbert space Cn to Cm, 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 with respect to an orthonormal basis.

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]