conjugate transpose
Definition If is a complex matrix, then theconjugate transpose is the matrix, where isthe complex conjugate
of , and is thetranspose
of .
It is clear that for real matrices, the conjugate transpose coincides withthe transpose.
0.0.1 Properties
- 1.
If and are complex matrices of same size, and are complex constants, then
- 2.
If and are complex matrices such that is defined, then
- 3.
If is a complex square matrix
, then
where and are the traceand the determinant
operators, and is the inverse operator.
- 4.
Suppose is the standard inner product on .Then for an arbitrary complex matrix ,and vectors , we have
Notes
The conjugate transpose of is also called the adjoint matrix of ,the Hermitian conjugate of (whence one usually writes ).The notation is also used for the conjugate transpose [2].In [1], is also called the tranjugate of .
References
- 1 H. Eves, Elementary Matrix
Theory, Dover publications, 1980.
- 2 M. C. Pease,Methods of Matrix Algebra, Academic Press, 1965.
See also
- •
Wikipedia,http://www.wikipedia.org/wiki/Conjugate_transposeconjugate transpose