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# Conjugate transpose

In mathematics, the conjugate transpose or adjoint 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
for 1 ≤ in and 1 ≤ jm.

 Table of contents 1 Example 2 Basic remarks 3 Properties of the conjugate transpose 4 Adjoint operator in Hilbert space

For example, if

then

## Basic remarks

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").

## Adjoint operator in Hilbert space

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.