Table of contents |
2 Basic remarks 3 Properties of the conjugate transpose 4 Adjoint operator in Hilbert space |
For example, if
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.
The square matrix A is called hermitian or self-adjoint if A = A*. It is called 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").
The final property given above shows that if one views A as a linear operator from the Euclidean Hilbert space Cn to Cm, 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.Example
then
Basic remarks
Properties of the conjugate transpose
Adjoint operator in Hilbert space