= Hermitian quadratic form
{c}
{title2=$q_T(x)=\langle Tx,x\rangle$}
A bounded <Hermitian operator> determines a <Hermitian form> $b_T(x,y)=\langle Tx,y\rangle$ and its real-valued diagonal expression $q_T(x)=b_T(x,x)$. On a complex <Hilbert space>,
$$
q_T(\alpha x)=|\alpha|^2q_T(x).
$$
Thus it is a <quadratic form> on the underlying real <vector space>; it is generally not a <quadratic form> over the complex <field>, whose scaling rule would involve $\alpha^2$. This distinction explains why replacing $x^*$ by $x^{\mathrm T}$ loses the geometry of a complex <inner product>.
For a finite-dimensional <Hermitian operator> with an <orthonormal basis> of <eigenvectors> $e_j$ and corresponding <eigenvalues> $\lambda_j$, the <spectral theorem> gives
$$
x=\sum_j c_je_j\quad\Longrightarrow\quad q_T(x)=\sum_j\lambda_j|c_j|^2.
$$
For a <unit vector>, this is an average of the <eigenvalues> with nonnegative weights summing to one. It is also the <Rayleigh quotient>, so its minimum and maximum are the smallest and largest <eigenvalues>.
In the <Hermitian operator> diagram, take $x(\theta)=(\cos\theta,\sin\theta)^{\mathrm T}$ for $D,P$, and $x_H(\theta)=\cos\theta\,e_1+\sin\theta\,e_3$ for $H$. The computed curves are
$$
q_D(x(\theta))=\cos(2\theta),\quad q_P(x(\theta))=\cos^2\theta,\quad q_H(x_H(\theta))=\cos^2\theta+3\sin^2\theta.
$$
These paths illustrate the extremal values; the <spectral theorem> supplies the bounds for every complex <unit vector>, including those outside the plotted paths.
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