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conjugate transpose
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(Definition)
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Definition If $A$ is a complex matrix, then the conjugate transpose $A^\ast$ is the matrix
, where $\bar{A}$ is the complex conjugate of $A$ , and
is the transpose of $A$ .
It is clear that for real matrices, the conjugate transpose coincides with the transpose.
- If $A$ and $B$ are complex matrices of same size, and $\alpha,\beta$ are complex constants, then \begin{eqnarray*} (\alpha A + \beta B)^\ast &=& \overline{\alpha} A^\ast + \overline{\beta} B^\ast,\\ A^{\ast\ast} &=& A. \end{eqnarray*}
- If $A$ and $B$ are complex matrices such that $AB$ is defined, then $$ (AB)^\ast = B^\ast A^\ast.$$
- If $A$ is a complex square matrix, then \begin{eqnarray*} \det (A^\ast) &=& \overline{ \det{A}}, \\ \operatorname{trace}(A^\ast) &=& \overline{ \operatorname{trace}{A}}, \\ (A^\ast)^{-1} &=& (A^{-1})^\ast, \end{eqnarray*}where $\operatorname{trace}$ and $\operatorname{det}$ are the trace and the determinant operators, and $^{-1}$ is the inverse
operator.
- Suppose $\langle \cdot, \cdot \rangle$ is the standard inner product on $\sC^n$ . Then for an arbitrary complex $n\times n$ matrix $A$ , and vectors $x,y\in \sC^n$ , we have $$ \langle Ax,y\rangle = \langle x,A^\ast y \rangle.$$
The conjugate transpose of $A$ is also called the adjoint matrix of $A$ , the Hermitian conjugate of $A$ (whence one usually writes
). The notation $A^\dagger$ is also used for the conjugate transpose [2]. In [1], $A^\ast$ is also called the tranjugate of $A$ .
- 1
- H. Eves, Elementary Matrix Theory, Dover publications, 1980.
- 2
- M. C. Pease, Methods of Matrix Algebra, Academic Press, 1965.
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"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 21411 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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Pending Errata and Addenda
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