Conjugate Transpose is Involution
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Theorem
Let $\mathbf A$ be a complex-valued matrix.
Let $\mathbf A^*$ denote the Hermitian conjugate of $\mathbf A$.
Then the operation of Hermitian conjugate is an involution:
- $\paren {\mathbf A^*}^* = \mathbf A$
Proof
\(\ds \sqbrk {\paren {\mathbf A^*}^* }_{i j}\) | \(=\) | \(\ds \overline {\sqbrk {\mathbf A^*}_{j i} }\) | Definition of Hermitian Conjugate | |||||||||||
\(\ds \) | \(=\) | \(\ds \overline {\paren {\overline {\sqbrk {\mathbf A}_{i j} } } }\) | Definition of Hermitian Conjugate | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqbrk {\mathbf A}_{i j}\) | Complex Conjugation is Involution |
So:
- $\paren {\mathbf A^*}^* = \mathbf A$
$\blacksquare$