Is a Matrix Zero if Its Inner Product with All Vector Combinations Equals Zero?

[bernoli123]

if A $$\in$$ C nxn,show that (x,Ay)=0 for all x,y $$\in$$ C[n],
then A=0

(x,Ay) denote standard inner product on C[n]

 

[chiro]

if A $$\in$$ C nxn,show that (x,Ay)=0 for all x,y $$\in$$ C[n],
then A=0

(x,Ay) denote standard inner product on C[n]

Have you tried using properties of norms, and axioms of the inner product (There is an axiom that deals with zero results)?

 

[HallsofIvy]

In particular, (e_i, Ae_i)= 0 for every member, $e_i$, of an orthonormal basis.