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
- <math>(A^*)[i,j] = \overline{A[j,i]}</math>
for 1≤
i≤
n and 1≤
j≤
m.
For example, if
- <math>A=\begin{bmatrix}3+i&2\\
2-2i&i\end{bmatrix}</math>
then
- <math>A^*=\begin{bmatrix}3-i&2+2i\\
2&-i\end{bmatrix}</math>
If the entries of A are real, then A^{*} coincides with the transpose A^{T} 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 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 last property 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[?].
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[?].
The adjoint matrix A^{*} should not be confused with the adjugate adj(A) (which in older texts is also sometimes called "adjoint").
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