On page 150 of Nocedal & Wright's Numerical Optimization (PDF), there is no proof of theorem 6.3. I was hoping for someone to point me to the proof or alternative provided it. The theorem is
Suppose $f: \mathbb R^n \rightarrow \mathbb R$ is strongly convex quadratic function $f(x) = 0.5x^TAx + b^Tx$ where $A$ is symmetric positive definite. Let $x_0$ be any starting point and $B_0$ be any symmetric positive definite matrix, and suppose that the matrices is $B_k$ are updated by the Broyden formula with $\phi_k \in [0,1]$. Define $\lambda_1 ^k\leq \lambda_2^k \leq \dots \leq \lambda_n^k $ the be the eigenvalues of the matrix $$A^{1/2} B_k^{-1}A^{1/2}$$
then for all $k$ we have $$\min\{\lambda_j^k,1\} \leq \lambda_j^{k+1} \leq \max\{\lambda_j^k,1\}$$ for $j=1,2,3,\dots,n$
Thank you, for your time and inputs, guys.