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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

$(A^*)[i,j] = \overline{A[j,i]}$
for 1≤in and 1≤jm.

For example, if

$A=\begin{bmatrix}3+i&2\\ 2-2i&i\end{bmatrix}$ then
$A^*=\begin{bmatrix}3-i&2+2i\\ 2&-i\end{bmatrix}$

If the entries of A are real, then A* coincides with the transpose AT of A.

This operation has the following properties:

• (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 Cn and any vector y in Cm. Here <.,.> denotes the ordinary Euclidean inner product (or dot product) on Cm and Cn.
The last property 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[?].

It is 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[?].