= Solution
A <Hermitian form> is a <sesquilinear form> $H:V\times V\to\mathbb C$ such that
$$
H(v,w)=\overline{H(w,v)}.
$$
Use the convention that $H$ is linear in its first argument and conjugate-linear in its second. Its <matrix of a Hermitian form>[matrix] in the basis $(v_1,\ldots,v_n)$ is
$$
A_{ij}=H(v_i,v_j).
$$
If $v=\sum_i x_iv_i$ and $w=\sum_jy_jv_j$, then
$$
H(v,w)=x^TA\overline y.
$$
Since $w_i=\sum_jp_{ij}v_j$, the coordinate row of each new basis vector is a row of $P$. Direct substitution gives
$$
H(w_i,w_k)=\sum_{j,l}p_{ij}A_{jl}\overline{p_{kl}},
$$
so the new matrix is
$$
\boxed{P A\overline P^{\,T}}.
$$
The subspace $N=V^\perp$ is the <radical of a bilinear form>[radical] of $H$. In coordinates it is the <kernel of a linear map>[kernel] of $A$, and the <rank-nullity theorem> gives
$$
\boxed{\dim N=n-\operatorname{rank}A}.
$$
Back to article page
