Let $V$ be a finite dimensional vector space and $S,T\in \mathcal L(V)$ where $\mathcal L(V)$ denotes the space of linear operators.
If $S,T$ are self-adjoint and have all eigen values positive show that $S+T$ has all eigen values non-negative.
I tried like this:
Let $\lambda $ be an eigen value of $S+T$ and let $(S+T)v=\lambda v$
Now since $S^{*}=S$ and $T^{*}=T\implies (S+T)^{*}=S+T$
Thus $\langle (S+T)v,(S+T)v\rangle =\langle \lambda v,\lambda v\rangle=|\lambda|^2\langle v,v\rangle$
Also
$\langle (S+T)v,(S+T)v\rangle=\langle v,(S+T)^2v\rangle$
How to show that $\lambda\ge 0 $ from above?
Will someone give some way to solve it?
Thank you