= Solution
A <Hermitian operator> satisfies the adjoint symmetry condition
$$
\langle\phi|\widehat A\psi\rangle=\langle\widehat A\phi|\psi\rangle
$$
for vectors in its specified domain. For a finite matrix this means $A=A^\dagger$. For an unbounded differential operator, boundary conditions and domains matter: symmetry on a chosen domain is not alone sufficient for <self-adjointness>. Physical observables require the corresponding <self-adjoint operator> and its spectral measure. This distinction prevents purely formal integration by parts from being mistaken for a complete operator definition.
\b[Real <eigenvalues>.] If $A|a\rangle=a|a\rangle$ with nonzero norm, Hermiticity implies
$$
a\langle a|a\rangle=\langle a|A|a\rangle
=\langle a|A|a\rangle^*=a^*\langle a|a\rangle,
$$
so $a=a^*$. Thus <eigenvalues> representing measurement outcomes are real.
\b[<Orthogonality>.] For eigenvectors $|a\rangle,|b\rangle$,
$$
a\langle b|a\rangle=\langle b|A|a\rangle
=\langle Ab|a\rangle=b\langle b|a\rangle.
$$
Hence $(a-b)\langle b|a\rangle=0$, and distinct <eigenvalues> have orthogonal eigenvectors. Within a degenerate eigenspace, <Gram-Schmidt orthogonalization> can choose an <orthonormal basis>. In finite dimensions the <spectral theorem> gives a complete orthonormal eigenbasis and
$$
\boxed{A=\sum_a aP_a,\qquad I=\sum_aP_a,}
$$
where $P_a$ projects onto each eigenspace. In infinite dimensions a continuous spectrum may require a spectral integral rather than a normalizable eigenvector for every possible outcome; a <position operator> on the full line is a familiar example.
\b[Measurements and probabilities.] In the discrete case the <Born rule> assigns <probability> $\langle\psi|P_a|\psi\rangle$ to the outcome $a$. For a nondegenerate eigenbasis, $|\psi\rangle=\sum_ac_a|a\rangle$ gives probabilities $|c_a|^2$, with
$$
\langle A\rangle=\sum_a a|c_a|^2\in\mathbb R,\qquad
(\Delta A)^2=\langle(A-\langle A\rangle)^2\rangle\ge0.
$$
The <variance> vanishes precisely when the state lies in an eigenspace with a single sharp outcome. A repeatable ideal projective measurement prepares the corresponding eigenspace; superpositions of different <eigenvalues> do not have a preassigned sharp value of that observable in this description.
\b[Compatibility and dynamics.] Commuting Hermitian matrices admit simultaneous diagonalization, permitting a common set of sharp labels. For example $L^2,L_z$ commute, whereas $L_x,L_y$ do not. In the differential-operator setting the corresponding spectral compatibility requires appropriate domain conditions. Noncommuting observables obey the <Robertson uncertainty principle>
$$
\boxed{\Delta A\,\Delta B\ge\frac12|\langle[A,B]\rangle|.}
$$
The <canonical commutation relation> then gives $\Delta x\Delta p\ge\hbar/2$.
The <Hamiltonian operator> represents <energy>. A self-adjoint $H$ generates <unitary time evolution>, $U(t)=e^{-iHt/\hbar}$, preserving norm and total <probability>; its <eigenvalues> determine stationary-state phases and transition energies. \b[Hermiticity supplies real outcomes and orthogonal eigenspaces, while self-adjointness and the spectral theorem make these measurement and evolution rules mathematically complete.]
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