Does \(v^\top A v = 0\) for all vectors v imply A is skew symmetric?

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jeng9rulemoreee jeng9rulemoreee

28-01-2017
Mathematics

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Does \(v^\top A v = 0\) for all vectors v imply A is skew symmetric?

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LammettHash LammettHash · Answer 1 · 2017-01-29T00:50:04+01:00

Let [tex]\mathbf v=\mathbf x+\mathbf y[/tex]. Then

[tex]0=\left(\mathbf v^\top\mathbf{Av}\right)^\top[/tex]
[tex]0=(\mathbf x+\mathbf y)^\top\mathbf A^\top(\mathbf x+\mathbf y)[/tex]
[tex]0=\mathbf x^\top\mathbf A^\top\mathbf x+\mathbf x^\top\mathbf A^\top\mathbf y+\mathbf y^\top\mathbf A^\top\mathbf x+\mathbf y^\top\mathbf A^\top\mathbf y[/tex]
[tex]0=\mathbf x^\top\mathbf A^\top\mathbf y+\mathbf y^\top\mathbf A^\top\mathbf x[/tex]
[tex]\mathbf x^\top\mathbf A^\top\mathbf y=-\mathbf y^\top\mathbf A^\top\mathbf x[/tex]
[tex]\mathbf x^\top\mathbf A^\top\mathbf y=-\left(\mathbf x^\top\mathbf{Ay}\right)^\top[/tex]

which requires [tex]\mathbf A=-\mathbf A^\top[/tex], so the answer is yes.