PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (198)
Orphanage
Unclass'd
Unproven (490)
Corrections (7)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
conjugate transpose (Definition)

Definition If $ A$ is a complex matrix, then the conjugate transpose $ A^\ast$ is the matrix $ A^\ast = \bar{A}\hspace{0.04cm} ^{\mbox{\scriptsize {T}}} \hspace{0.02cm}$, where $ \bar{A}$ is the complex conjugate of $ A$, and $ A\hspace{0.04cm} ^{\mbox{\scriptsize {T}}} \hspace{0.02cm}$ is the transpose of $ A$.

It is clear that for real matrices, the conjugate transpose coincides with the transpose.

Properties

  1. If $ A$ and $ B$ are complex matrices of same size, and $ \alpha,\beta$ are complex constants, then
    $\displaystyle (\alpha A + \beta B)^\ast$ $\displaystyle =$ $\displaystyle \overline{\alpha} A^\ast + \overline{\beta} B^\ast,$  
    $\displaystyle A^{\ast\ast}$ $\displaystyle =$ $\displaystyle A.$  

  2. If $ A$ and $ B$ are complex matrices such that $ AB$ is defined, then
    $\displaystyle (AB)^\ast = B^\ast A^\ast.$
  3. If $ A$ is a complex square matrix, then
    $\displaystyle \det (A^\ast)$ $\displaystyle =$ $\displaystyle \overline{ \det{A}},$  
    $\displaystyle \operatorname{trace}(A^\ast)$ $\displaystyle =$ $\displaystyle \overline{ \operatorname{trace}{A}},$  
    $\displaystyle (A^\ast)^{-1}$ $\displaystyle =$ $\displaystyle (A^{-1})^\ast,$  

    where $ \operatorname{trace}$ and $ \operatorname{det}$ are the trace and the determinant operators, and $ ^{-1}$ is the inverse operator.
  4. Suppose $ \langle \cdot, \cdot \rangle$ is the standard inner product on $ \mathbb{C}^n$. Then for an arbitrary complex $ n\times n$ matrix $ A$, and vectors $ x,y\in \mathbb{C}^n$, we have
    $\displaystyle \langle Ax,y\rangle = \langle x,A^\ast y \rangle.$

Notes

The conjugate transpose of $ A$ is also called the adjoint matrix of $ A$, the Hermitian conjugate of $ A$ (whence one usually writes $ A^\ast = A\hspace{0.04cm} ^{\mbox{\scriptsize {H}}} \hspace{0.02cm}$). The notation $ A^\dagger$ is also used for the conjugate transpose [2]. In [1], $ A^\ast$ is also called the tranjugate of $ A$.

References

1
H. Eves, Elementary Matrix Theory, Dover publications, 1980.
2
M. C. Pease, Methods of Matrix Algebra, Academic Press, 1965.

See also



"conjugate transpose" is owned by Koro. [ full author list (2) | owner history (1) ]
(view preamble)

View style:

See Also: transpose

Other names:  adjoint matrix, Hermitian conjugate, tranjugate
Log in to rate this entry.
(view current ratings)

Cross-references: vectors, inner product, inverse, operators, determinant, trace, square matrix, size, real, clear, transpose, complex conjugate, matrix, complex
There are 22 references to this entry.

This is version 7 of conjugate transpose, born on 2003-06-21, modified 2006-09-13.
Object id is 4382, canonical name is ConjugateTranspose.
Accessed 17320 times total.

Classification:
AMS MSC: 15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
 15A15 (Linear and multilinear algebra; matrix theory :: Determinants, permanents, other special matrix functions)
Pending Errata and Addenda
None.
[ View all 5 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)