Linear-algebra lemmas (PSD operators)
A grab-bag of facts used repeatedly in spectral arguments.
PSD decomposition
If $A=B^{\mathsf T}B$ then $\langle Ax,x\rangle=\|Bx\|^2$ and $\ker A=\ker B$.
Monotonicity
If $A\preceq C$ (i.e., $C-A\succeq 0$), then $\lambda_i(A)\le \lambda_i(C)$ for all $i$ (ordered eigenvalues with multiplicity).
Kernel inclusion trick
If $A\preceq C$ and $\ker(C)\subseteq\ker(A)$, then $(\ker A)^\perp \subseteq (\ker C)^\perp$ and $$\lambda_{\min}^+(C)\ \ge\ \lambda_{\min}^+(A).$$
The hypothesis $\ker(C)\subseteq\ker(A)$ is the point: without it, comparing “smallest positive” can be subtle.
Sherman–Morrison (invertible case)
If $A$ is invertible and $C=A+uu^{\mathsf T}$ then $$C^{-1}=A^{-1} - \frac{A^{-1}uu^{\mathsf T}A^{-1}}{1+u^{\mathsf T}A^{-1}u}.$$ This is useful for tracking resolvents or effective resistances; for PSD operators with kernel, one uses pseudoinverses.