Conjugate transpose

In mathematics, the conjugate 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. Formally

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

for 1 ≤ in and 1 ≤ jm.

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^*, A^H, \mbox{ or } A^\dagger\,. <math>




2-2i&i\end{bmatrix}<math> then



Basic remarks

If the entries of A are real, then A* coincides with the transpose AT 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

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 in older texts is also sometimes called "adjoint").

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 Cm. Here <.,.> denotes the ordinary Euclidean inner product (or dot product) on Cm and Cn.


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

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.

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