$x^TAx=0$ for all $x$ when $A$ is a skew symmetric matrix

$\begingroup$

Let $A$ be an $n\times n$ skew symmetric matrix.
Show that $x^TAx =0 \ \forall x \in \mathbb R^n$.

How to prove this?

$\endgroup$ 0

2 Answers

$\begingroup$

You know that $-A=A^T$, so

$x^T A x = (x, Ax)$ (1)

but we also have

$x^T A x = x^T (-A^T) x = -x^T A^T x = -(Ax)^Tx = -(Ax,x)$ (2)

Now notice (1) and (2) need to be the same.

$\endgroup$ 2 $\begingroup$

Hint$$(x^TAx)^T=x^TA^Tx=-x^TAx$$ and $x^TAx$ is a real(Why?) so...

$\endgroup$