In mathematics, the conjugate transpose, also known as the Hermitian transpose complex matrix matrix obtained by transposing and applying complex conjugation,,.

For real.

Definition

is formally defined by

, and the overbar denotes a scalar complex conjugate.

This definition can also be written as

denotes the matrix with complex conjugated entries.

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

• , commonly used in linear algebra

• , commonly used in linear algebra

• (sometimes pronounced as A dagger), commonly used in quantum mechanics

• , although this symbol is more commonly used for the Moore–Penrose pseudoinverse

denotes the matrix with only complex conjugated entries and no transposition.

Example

.

We first transpose the matrix:

Then we conjugate every entry of the matrix:

Basic remarks

is called

• Hermitian or self-adjoint.

• Skew Hermitian.

• Normal.

• Unitary.

are both Hermitian and in fact positive semi-definite matrices.

should not be confused with the adjugate, which is also sometimes called adjoint.

real matrices, obeying matrix addition

That is, denoting each complex by the real matrix of the linear transformation on the Argand diagram (viewed as the real), affected by complex .

matrix made up of complex numbers.

as the rotation matrix

The complex conjugate) is encoded as the matrix transpose.

Properties

• of the same dimensions.

• .

• . Note that the order of the factors is reversed.

• , i.e. Hermitian transposition is an involution.

• denotes the determinant .

• denotes the trace.

• is invertible if and only if.

• The eigenvalues are the complex conjugates of the eigenvalues.

• denotes the standard complex inner product.

Generalizations

as a linear transformation from Hilbert space corresponds to the adjoint operator. 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.

is a linear map from a complex vector space, then the complex conjugate linear map as well as the transposed linear map. It maps the conjugate dual.

See also

• Complex dot product

• Hermitian adjoint

• Adjugate matrix

References

External links