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For a paper, I am looking for a reference in the literature for the following theorem:

Given a symmetric non degenerate bilinear form $\phi$ on a finite dimensional vector space 𝑉, with $\dim 𝑉 >1$, and $\phi(e_i, e_j)=0$ if $i+j>n+1$, there is a basis of $𝑒_1$,...,$𝑒_𝑛$ of V such that the matrix 𝐵 associated to $\phi$ with respect to $𝑒_1$,...,$𝑒_𝑛$ is the anti-diagonal matrix.

A proof for this statement is given at

Anti-diagonal matrix symmetric bilinear form

(though the original poster left out an important condition)

Thank you in advance

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  • $\begingroup$ What do you mean by anti-diagonal? Don't you mean diagonal? $\endgroup$ May 16, 2023 at 20:50
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    $\begingroup$ It is false. There is no basis of $\mathbb R^2$ such that $\begin{bmatrix}0&a\\a&0\end{bmatrix}$ represents the Euclidean dot product. $\endgroup$
    – mr_e_man
    May 16, 2023 at 20:50
  • $\begingroup$ Sorry, corrected the post to indicate that the values of the form below the anti-diagonal are all zero. $\endgroup$ May 17, 2023 at 18:29

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