Proof of Identity including matrices and vectors

Question

Proof of $H_+=H_b$ with $$H_+=H+\rho((s-Hy)s^T+s(s-Hy)^T)-\rho^2(s-Hy)^Tyss^T$$ $$H_b=(I-\rho sy^T)H(I-\rho ys^T)+\rho ssT$$ $H\in Mat_n(\mathbb R)$ symmetric and pos.definit; $s,y\in\mathbb R^n, \rho=\frac{1}{y^Ts}\gt0$.

Start: $$H_+=H+\rho ss^T-\rho Hys^T+\rho ss^T-\rho sy^TH-\rho^2s^Tyss^T+\rho^2 y^THyss^T$$$$=H+\rho ss^T-\rho Hys^T-\rho sy^TH+\rho^2 y^THyss^T$$$$H_b=H-\rho sy^TH-\rho Hys^T+\rho^2 sy^THys^T+\rho ss^T$$ So when I compare $H_+ and H_b$ I get that $$\rho^2 sy^THys^T=\rho^2 y^THyss^T$$ but I dont know how to proof this.

Answer 1

\begin{equation} \rho^2sy^THys^T=\rho^2y^THyss^T\\sy^THys^T=y^THyss^T \end{equation}

but $y^THy\in\mathbb{R}$ because H defines a scalar product so $y^THy$ is the squared norm of $y$. So we have:

\begin{equation} ss^T=ss^T. \end{equation}