Table of contents |

2 Basic remarks 3 Properties of the conjugate transpose 4 Adjoint operator in Hilbert space |

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.

The square matrix *A* is called hermitian or self-adjoint if *A* = *A*^{*}. It is called normal if *A ^{*}A* =

Even if *A* is not square, the two matrices *A ^{*}A* and

The adjoint matrix *A*^{*} should not be confused with the adjugate adj(*A*) (which in older texts is also sometimes called "adjoint").

- (
*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*. - (
*A*^{*})^{*}=*A*for any matrix*A*. - <
*Ax*,*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 ordinary Euclidean inner product (or dot product) on**C**^{m}and**C**^{n}.

The final property given above shows that if one views *A* as a linear operator from the Euclidean Hilbert space **C**^{n} to **C**^{m}, then the matrix *A*^{*} corresponds to the **adjoint operator**.

In fact it can be used to define what is meant by that. Assuming now we are in a Hilbert space *H*, the relation

<*Ax*,*y*> = <*x*, *A*^{*}*y*>

can be used to define the *adjoint operator* *A*^{*}, by means of the Riesz representation theorem.