2nd half of linear algebra proof

[Steven G]

Let A be an nxn matrix
Suppose x^TAx =0 for all n-vectors x. I want to conclude that A= -A^T.

Here is what I have so far.

Let x and y be 2 n-vectors

0 = (x+y)^TA(x+y) = x^TAx + y^TAy + x^TAy + y^TAx
The two bold terms are both 0. So x^TAy + y^TAx = 0 or x^TAy = -y^TAx = {-y^TAx}^T = -x^TA^Ty =x^T(-A^T)y,

That is x^TAy = x^T(-A^T)y

I just can't justify to myself why I can now say that A=-A^T

Please help. Thanks!

 

[Steven G]

Let A be an nxn matrix
Suppose x^TAx =0 for all n-vectors x. I want to conclude that A= -A^T.

Here is what I have so far.

Let x and y be 2 n-vectors

0 = (x+y)^TA(x+y) = x^TAx + y^TAy + x^TAy + y^TAx
The two bold terms are both 0. So x^TAy + y^TAx = 0 or x^TAy = -y^TAx = {-y^TAx}^T = -x^TA^Ty =x^T(-A^T)y,

That is x^TAy = x^T(-A^T)y

I just can't justify to myself why I can now say that A=-A^T

Please help. Thanks!

Got it!
x^TAy + x^T(A^T)y =0

Then use x^T(A+A^T)y = 0 for all x,y ....