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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) ]
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See Also: transpose

Other names:  adjoint matrix, Hermitian conjugate, tranjugate
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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 17649 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)
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