Therefore, P is a Hermitian matrix.
Also
Therefore, Q is also a Hermitian matrix.
Thus, A can be expressed in the form (1).
For A to be unique, let A = R + iS, where R and S are both Hermitian matrices. We have
Aθ = (R + iS)θ = Rθ + (iS)θ = Rθ + iSθ = Rθ - iSθ
= R - iS (since R and S are both Hermitian matrices)
Therefore,
A + Aθ = (R + iS) + (R - iS) = 2R
⇒ R = 1/2(A + Aθ) = P
Also,
A - Aθ = (R + iS) - (R - iS) = 2iS
⇒ S = 1/2i(A - Aθ) = Q
Hence, expression (1) for A is unique