Matrix Symmetry Proof

• May 8th 2011, 03:22 PM
StaryNight
Matrix Symmetry Proof
I am to prove that if A is antisymmetric, then x^TAx = 0 for any column vector x. I know that by definition A^T = -A, so (x^TAx) = -(x^TAx). I have tried taking the transpose of both sides but this does not seem to help prove the above.

Any help would be appreciated.
• May 8th 2011, 04:12 PM
FernandoRevilla
Use that

$x^tAx\in \mathbb{R}^{1\times 1}$

so, it is symmetric.
• May 8th 2011, 04:43 PM
StaryNight
Quote:

Originally Posted by FernandoRevilla
Use that

$x^tAx\in \mathbb{R}^{1\times 1}$

so, it is symmetric.

This completes the proof very nicely. However, how do I show formally that this is true?
• May 8th 2011, 05:22 PM
Deveno
xᵀ is 1xn, A is nxn, x is nx1, so Ax is nx1, so xᵀ(Ax) is 1x1 (it's just the inner product of x and Ax).