I want to show: If a matrix $A$ is skew symmetric, that is if $A^t=-A$ then $x^tAx=0$ for all vectors $x$.
Please give hints and thought processes for the proof. I am quite stuck on this question. Please answer the following question:
- In general, when you are proving an algebraic statement, how do you get intuition? A statement like this means nothing to me other than manipulating a bunch of symbols. This sounds like a "strategy" setup for failure…
Best Answer
$$x^TAx=d\to(x^TAx)^T=d^T=d\to x^TAx=-d\Rightarrow\begin{cases} x^TAx=d \\ x^TAx=-d \\ \end{cases}\color{red}{\Rightarrow}2(x^TAx)=0\to x^TAx=0$$