[[Matrix algebra over a field]]
# Conjugate transpose

The **conjugate transpose** or **Hermitian conjugate** $A^{\dagger}$ of a complex matrix $A$
is the [[Matrix transpose]] $\tp A$ with [[complex conjugate]] entries. #m/def/linalg 
Thus if $A = (a_{ij})$ then $A^{\dagger} = (\overline{a_{ji}})$.
The conjugate transpose is an important manifestation of [[Duality]]:
If vectors in a complex vector space $\mathbb{C}^n$ are represented as column vectors,
the [[Dual space]] is that of row vectors
and the duals of linear maps and vectors with respect to the standard [[Inner product space|inner product]] is given by the conjugate transpose.

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