# Derive Hessian inverse update using Sherman-Morrison in Quasi Newton Method

From Nocedal's Numerical Optimization book, the Hessian approximation equation is given using Symmetric Rank 1 (SR1) formula as follows:

$$B_{k+1} = B_k + \frac{(y_k - B_ks_k)(y_k - B_ks_k)}{(y_k - B_ks_k)^Ts_k}^T$$

where $$y_k = \nabla f(x_{k+1}) - \nabla f(x_k)$$ and $$sk = x_{k+1} - x_k$$

It further states that using Sherman-Morrison formula (Consider $$\bar A = A + ab^T$$, then $$A^{-1} = \bar A^{-1} - \frac {A^{-1}ab^TA^{-1}}{1+b^TA^{-1}a}$$ for $$\bar A$$ matrix)

The equation of inverse of Hessian can be derived using the above formula as follows:

$$H_{k+1} = H_k + \frac{(s_k - H_ky_k)(s_k - H_ky_k)^T}{(s_k - H_ky_k)y_k}$$

The problem is that to me that derivation is not obvious and I've tried solving this algebrically but I'm unable to get the derivation right from $$H_{k+1}$$ by manipulating $$B_{k+1}$$.