I saw in an article the flowing inequality for vectors $ x,y\in R^n , Q \in R^{n \times n} $ : $$ x^TQy\le {\frac 12}\lambda_{max}(Q){\Vert x \Vert}^2+{\frac 12}\lambda_{max}(Q){\Vert y \Vert}^2 $$ or some other version of it:
$$ x^TQy\le {\frac 1{2k^2}}\lambda_{max}(Q){\Vert x \Vert}^2+{\frac {k^2}2}\lambda_{max}(Q){\Vert y \Vert}^2 $$ I have two questions:
- how can i prove the inequality?
- is the a minimum version of it like: $$ {\frac 1{2k^2}}\lambda_{min}(Q){\Vert x \Vert}^2+{\frac {k^2}2}\lambda_{min}(Q){\Vert y \Vert}^2\le x^TQy $$ or $$ {\frac 1{2}}\lambda_{min}(Q){\Vert x \Vert}^2+{\frac {1}2}\lambda_{min}(Q){\Vert y \Vert}^2\le x^TQy $$