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

${\displaystyle (A^{*})[i,j]={\overline {A[j,i]}}}$

for 1 ≤ i ≤ n and 1 ≤ j ≤ m.

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:

${\displaystyle A^{*},A^{H},{\mbox{ or }}A^{\dagger }\,.}$

Example

If

${\displaystyle A={\begin{bmatrix}3+i&2\\2-2i&i\end{bmatrix}}}$ 

then

${\displaystyle A^{*}={\begin{bmatrix}3-i&2+2i\\2&-i\end{bmatrix}}.}$ 

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

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.

External links

• Conjugate transpose on Mathworld, Wolfram research
• "Conjugate transpose". PlanetMath.