Puneet Varma (Editor)

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 (i.e., negating their imaginary parts but not their real parts). The conjugate transpose is formally defined by

Contents

( A ∗ ) i j = A j i ¯

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

This definition can also be written as

A ∗ = ( A ¯ ) T = A T ¯

where A T denotes the transpose and A ¯ 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:

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

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

    Example

    If

    A = [ 1 − 2 − i 1 + i i ]

    then

    A ∗ = [ 1 1 − i − 2 + i − i ]

    Basic remarks

    A square matrix A with entries a i j is called

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

    • 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".

    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

    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 + i b ≡ ( a − b b a ) .

    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 R 2 ) affected by complex z-multiplication on 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

  • (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)∗ = 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 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.
  • ⟨ A x , y ⟩ = ⟨ x , A ∗ y ⟩ for any m-by-n matrix A, any vector x in C n and any vector y in C m . Here, ⟨ ⋅ , ⋅ ⟩ denotes the standard complex inner product on C m and C n .

    • Generalizations

    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.

    References

    Conjugate transpose Wikipedia
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