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What are the differences between $^t$, $^\dagger$, $^*$, $^H$, and $^T$. Further, to which is which associated, and are there some cases where one is used for both?

I'd like to see definitions for each group of superscript symbols and how $T$ the transformation and $A$ the matrix are related. I feel it would be good to have this as an answer here for all eyes to see.

  • $\begingroup$ I thing generally $t$ and $T$ are for transpose, $*$ and $H$ are for the conjugate transpose or adjoint, and the dagger for the pseudo-inverse. $\endgroup$ – copper.hat Jun 4 '13 at 3:05
  • $\begingroup$ Is what you're saying is that $A^*$ and $T^*$ convey the same information? $\endgroup$ – Tyquan Pesik Jun 4 '13 at 4:13
  • $\begingroup$ Don't forget ${}^t A$. $\endgroup$ – GEdgar Jun 4 '13 at 12:01

When $A$ is a matrix,

  • $A^t$ and $A^T$ stand for the transpose
  • $A^*$ (and possibly $A^H$, but I don't remember seeing it used this way) stand for Hermitian adjoint. The Wikipedia article says that $\dagger$ is also used for this purpose, by the kind of people who say bra and ket.
  • $A^\dagger$ stands for the pseudoinverse
  • 1
    $\begingroup$ I have seen people using $A^{*}$ for complex conjugate of $A$ and $A^{\dagger}$ for Hermitian adjoint. They even appear in same article. What is what is usually clear in the context. $\endgroup$ – achille hui Jun 4 '13 at 3:10
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    $\begingroup$ For pseudoinverse, $A^\dagger$ was popular in earlier linear algebra literature, but $A^+$ seems to be more widespread in recent years. Also, some physicists seem to use $A^\dagger$ for Hermitian adjoint. Transposes can be denoted by $A^T$ (most common), $A^t,\,A^\top$ and $\phantom{}^tA$. $\endgroup$ – user1551 Jun 4 '13 at 3:48
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    $\begingroup$ ... and $A^\prime$. (A habit initiated by Matlab-programmers?) $\endgroup$ – Myself Jun 4 '13 at 4:37
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    $\begingroup$ You really want $*$ for the adjoint, because if $H$ is a complex Hilbert space, then $B(H)$ is a $C^*$-algebra where the star operation is precisely "take the adjoint". $\endgroup$ – kahen Jun 4 '13 at 10:46

If you use $A^\sf T$ and $A^\sf H$ respectively for the transpose and Hermitian conjugate of $A$ (which I prefer), then you are free to use $t,T,H,$ and * for other things, which is useful. The notation $A^+$ for the pseudo-inverse is aesthetically in line with this and more modern than $A^\dagger$.


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