Consider
- $V$ a finite dimensional real vector space with Euclidean metric $\langle|\rangle$
- $\langle|\rangle_{\mathbb{C}}$ the induced Hermitian metric on the complexification $V_{\mathbb{C}}=V\otimes\mathbb{C}$
- $f(A)$ the complex-linear extension of any real-linear map $A:V\longrightarrow V$
- $A’$ such that $\langle Av_1|v_2\rangle=\langle v_1|A’v_2\rangle$ for all $v_1,v_2\in V$
- $f(A)’$ such that $\langle f(A)w_1|w_2\rangle_{\mathbb{C}}=\langle w_1|f(A)’w_2\rangle_{\mathbb{C}}$ for $w_1,w_2\in V_{\mathbb{C}}$
I need to show that $f(A’)=f(A)’$. Is it enough to show that $\langle f(A)w_1|w_2\rangle_{\mathbb{C}}=\langle w_1|f(A’)w_2\rangle_{\mathbb{C}}$? How can I do that? Thanks for your help !