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what is wrong about this proof?

If [tex]x^TAx=0[/tex] then A is antisymetric matrix. True? false?

P: False

[tex]A=-A^T[/tex]

[tex]x^TAx=-x^TA^Tx[/tex]

[tex]x^TAx=-(Ax)^Tx[/tex]

[tex]x^TAx=-\lambda\Vert x\Vert^2[/tex]

If x^T.A.x is zero, then must be [tex]-\lambda\Vert x\Vert^2[/tex], but ||x|| is real nonzero number and lambda must be zero. But antisymetric matrix has imaginary eigenvalues [tex]b\mathrm{i}[/tex], and 0 is not in this form. So