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The characteristic polynomial factors as $\det (\lambda I-A)=(\lambda -1)({\lambda }^2-1)$. Thus $\boxed{\lambda =1\text{ twice},\lambda =-1\text{ once}}$ in algebraic multiplicity. An orthonormal basis of corresponding eigenvectors isDirect multiplication verifies $A{v}_1=v_1$, $A{v}_2=v_2$, $A{v}_3=-v_3$. Their mutual Hermitian inner products vanish and each has norm one, which proves linear independence. For the repeated eigenvalue, Gaussian elimination givesHence $\boxed{\mathrm{rank}(A-I)=1,\dim \ker (A-I)=2}$. The second equality follows from the rank-nullity theorem and agrees with the two-dimensional eigenspace. A repeated eigenvalue alone would not imply a two-dimensional eigenspace for an arbitrary matrix; here the Hermitian matrix is diagonalizable.

The characteristic equation of the damped harmonic oscillator is $r^2+\gamma r+{\omega }_0^2=0$. In the underdamped case its roots are $-\gamma /2\pm i{\omega }_d$, where ${\omega }_d=\sqrt{{\omega }_0^2-{\gamma }^2/4}$. Thus the exact real solution is $y=A{e}^{-\gamma t/2}\cos ({\omega }_{d}t+\alpha )$. For $\gamma \ll {\omega }_0$, ${\omega }_d={\omega }_0+O({\gamma }^2/{\omega }_0)$, giving the requested leading approximationFor mass $m$ and spring constant $m{\omega }_0^2$, the mechanical energy is $E=\frac{1}{2}m({\dot{y}}^2+{\omega }_0^2{y}^2)$. Differentiating and using the differential equation gives $\dot{E}=-m\gamma {\dot{y}}^2$. In particular, energy decay of a viscously damped harmonic oscillator is not an exact pointwise exponential decay: $\dot{E}=0$ at a turning point. To leading order in $\gamma /{\omega }_0$, the period average isThe quality factor is $Q=2\pi \overline{E}/(\text{energy lost in one cycle})$. Using the period $T\simeq 2\pi /{\omega }_0$ and the small fractional loss $\gamma T$ gives $\boxed{Q\simeq {\omega }_0/\gamma }$. Equivalently, the energy-decay definition $Q={\omega }_0\overline{E}/{\overline{P}}_{\mathrm{loss}}$ yields the same leading result. Replacing ${\omega }_d$ by ${\omega }_0$ in the oscillatory phase is not uniform over arbitrarily long times; its phase error is $O({\gamma }^{2}t/{\omega }_0)$.