conjugate transpose
Definition If A is a complex matrix, then the
conjugate transpose A∗ is the matrix
A∗=ˉAT, where ˉA is
the complex conjugate
of A, and AT is the
transpose
of A.
It is clear that for real matrices, the conjugate transpose coincides with the transpose.
0.0.1 Properties
-
1.
If A and B are complex matrices of same size, and α,β are complex constants, then
(αA+βB)∗ = ˉαA∗+ˉβB∗, A∗∗ = A. -
2.
If A and B are complex matrices such that AB is defined, then
(AB)∗=B∗A∗. -
3.
If A is a complex square matrix
, then
where and are the trace and 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
Title | conjugate transpose |
---|---|
Canonical name | ConjugateTranspose |
Date of creation | 2013-03-22 13:42:18 |
Last modified on | 2013-03-22 13:42:18 |
Owner | Koro (127) |
Last modified by | Koro (127) |
Numerical id | 10 |
Author | Koro (127) |
Entry type | Definition |
Classification | msc 15-00 |
Classification | msc 15A15 |
Synonym | adjoint matrix |
Synonym | Hermitian conjugate |
Synonym | tranjugate |
Related topic | Transpose |