A max-min inequality for a rank-1 perturbed symmetric matrix Can someone explain why the underlined inequality is true? Thank you for your time. ![enter image description here](

I just realized that this is just a consequence of the Courant-Fischer theorem for symmetric matrices which states that $\substack{\text{max}\\\\\text{dim}(L)=k}\substack{\text{min}\\\x\in L\\\x\
eq0}\dfrac{x^\text{T}Ax}{x^\text{T}x}=\lambda_k(A)$ with the eigenvalues ordered as $\lambda_1(A)\geq\lambda_2(A)\geq\cdots\geq\lambda_n(A).$