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For any real $n \times n$ skew-symmetric matrix $A,$ show there exists an orthogonal matrix $P$ such that $$PAP^T = \begin{pmatrix}{} \Lambda_1 & \\ & \Lambda_2 & \\ & & \ddots & \\ & & & \Lambda_\frac{n}{2} \end{pmatrix},$$ where $\Lambda_i = \begin{pmatrix}{} 0 & \lambda_i \\ -\lambda_i & 0 \\ \end{pmatrix}$, $\lambda_i \geq 0.$

I have found a decent proof of the above, but unfortunately it uses notions of linear (skew) transformations. I am having difficulty locating a proof that uses bilinear forms instead and I would like to try and understand that approach.

user26857
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Abdul Miah
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  • I think it very likely that the algorithm I asked about at link below can be adapted to skew symmetric matrices, with all matrix entires used rational, if the original is rational. http://math.stackexchange.com/questions/1388421/reference-for-linear-algebra-books-that-teach-reverse-hermite-method-for-symmetr I would not expect to finish such a computer program today. The one for symmetric (integer) matrices took me six months, on and off. – Will Jagy Apr 26 '18 at 18:15
  • Oh, you want orthogonal.....If you can prove that, I cannot do any better. My change of variable matrices are determinant $\pm 1,$ but definitely not orthogonal. The whole point of that method is to keep all numbers used in the smallest field containing the entries of the beginning matrix. – Will Jagy Apr 26 '18 at 18:18
  • Cf. https://math.stackexchange.com/q/1682385/96384, https://math.stackexchange.com/q/2719021/96384, https://math.stackexchange.com/q/388750/96384. – Torsten Schoeneberg Dec 31 '20 at 18:43
  • Hi Abdul! Where did you find the proof using linear (skew) transformations? How exactly does one find this matrix P? – Whoopy Jul 17 '22 at 23:20

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