Conjugate Transpose is Involution

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:

$\paren {\mathbf A^*}^* = \mathbf A$

$\blacksquare$