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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


Basic remarks

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").

Properties of the conjugate transpose

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.