I've been given this as an assignment:
"From definitions of different classes of matrices, prove the following claims:
A) Positive => Symmetric…"
There is also a hint: If $<Au,u> = <Bu,u>$ for all $u$, then $A=B$.
The definitions have been given: A matrix is symmetric if $A^*=A$ and positive if $<Au,u>$ is positive for all $u$. The adjugate $A^*$ is defined as the matrix which satisfies $<Ax,y>=<x,A^*y>$.
I tried this:
The inner product is commutative, so:
So using the hint:
But for this proof, I didn't need the matrix $A$ to be positive. I seem to have proven that every matrix is symmetric. This probably isn't true, so what was my mistake?