# Matrix semi-definite positive

I have to show that given $$G = A^{T}A$$ and $$E = B^{T}B$$. the matrix $$G$$ or $$E$$ is semi-definite positive and also $$G + E$$. Do you think this is correct?
Because the dot product is a definite positive bilinear form, we can show that: $$\langle v,Gv\rangle = \langle v,A^{T}Av\rangle = \langle Av,Av\rangle \geq 0$$ And for $$E + G$$ we have using the formula above: $$\langle (G+E)v,v\rangle = \langle Gv+Ev,v\rangle = \langle Gv,v\rangle + \langle Ev,v\rangle \geq 0$$

When dealing with finite dimensional spaces , using the matrix notation $$ == u^{T}v$$ can make things a bit more clear.
• Using the properly-typeset matrix notation $\langle u, v\rangle = u^T v$ will make things even more clear than that. – Misha Lavrov Oct 8 '18 at 19:51