Transpose of conjugate transpose?

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What is the transpose of a conjugate transpose? Let's say we have a matrix $A$ and its conjugate transpose $A^{*}$, would the transpose of the conjugate transpose be the conjugate of $A$, i.e. $\bar{A}$?

Would one represent that as follows:

$A^{*T} = \bar{A}$

I have seen this written out as numpy code, but I don't know how to represent it.

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1 Answer

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The transpose of the transpose is the original matrix, as transposition is an involution.

So you're right, the transpose of the conjugate transpose of a matrix $A$ is just the conjugate $\bar A$, although I think this notation is not heavily used in linear algebra.

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