Conjugate transpose

In linear algebra we obtain the adjoint matrix by conjugating all entries of a complex matrix and the matrix then transposed. The adjoint matrix satisfies certain Vertauschungsbedingungen for scalar products.

Other spellings for the adjoint matrix, and. However, the notation is ambiguous because it is also used for the adjoint or complementary matrix.

Definition

Let be a matrix over the field of real or complex numbers, is called or the.

The adjoint matrix is to be defined by the following property:

Here, the standard scalar called the.

Calculation and calculation rules

Is a real matrix, the adjoint matrix to the transpose of:

Is a complex matrix, then the to adjoint matrix is the transpose of the complex conjugate of:

Applies so called self-adjoint. In the real case, the matrix then is also called symmetric and Hermitian in the complex case also.

Below are and matrices, and a complex number, then:

Generalization

In the functional analysis, the adjoint matrix is generalized to the adjoint operator.

For a morphism between two Hilbert spaces is an adjoint morphism by the property:

Defined. One can then make a connection to the dual operator.

• Matrix