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?
Linear Algebra – Prove x^TAx=0 for Skew Symmetric Matrix A
linear algebramatrices
linear algebramatrices
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?
Best Answer
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.