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 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[?].
The adjoint matrix A* should not be confused with the adjugate adj(A) (which in older texts is also sometimes called "adjoint").