Trace of matrices

Question

Well, I was solving exercises and here is one I cant solve.

$tr(A^TA)tr(B^TB)\geq tr(A^TB)^2$

I have tried it and I come with proof that $tr(A^TA)>0$ I dont know how it helps.

Thank you for your help.

Answer 1

For any square or rectangular matrx $C,$ we find $$ \operatorname{trace} C^T C \geq 0$$ because it is simply the sum of the squares of all elements.

Next, for any square matrix $M$ we have $$ \operatorname{trace} M^T = \operatorname{trace} M$$ Detail, $$(G^T H )^T = H^T G $$

Finally, $$ \operatorname{trace} \; (A-B)^T (A-B) \geq 0. $$ Expand this, there is a factor of $2$ you are missing.