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My question is over the finite field $\mathbf{F}_q$ of $q$ elements. It is well known that a symmetric matrix of odd rank corresponds to a parabolic quadratic form but even rank symmetric matrices correspond to either elliptic or hyperbolic quadratic forms. The number of symmetric matrices of fixed rank over $\mathbf{F}_q$ is known. For a given $m$, is the number of symmetric matrices of rank $2r+1$ and size $m\times m$ that correspond to elliptic and hyperbolic quadratic form known? If yes, what would be a suitable reference to understand it?

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    $\begingroup$ Number of quadratic forms up to what? If you mean up to isometry, there are essentially two nonisometric quadratic form of a given dimension over a finite field of characteristic different from 2. The reason is because over such fields the quadratic forms are classified by their dimension and their discriminant. Their discriminant has two values, hence the number is 2. $\endgroup$ – Name Jun 19 '18 at 13:43
  • $\begingroup$ Thank you Name but I am interested in the number of symmetric matrices of fixed rank that correspond to hyperbolic(elliptic) quadratic form. For example, there are $\frac{q(q^2-1)}{2}$ many $2\times 2$ matrices of rank 2 are of one type and remaining $\frac{q(q-1)^2}{2}$ many are of other types. $\endgroup$ – Singh Jun 19 '18 at 14:00
  • $\begingroup$ I have found a reference that answers this question. For a detailed solution, one can refer to "REPRESENTATIONS BY QUADRATIC FORMS IN A FINITE FIELD" by L. CARLITZ. $\endgroup$ – Singh Jun 21 '18 at 11:43
  • $\begingroup$ What is the definition of an elliptic or hyperbolic quadratic form over a finite (non-ordered) field? $\endgroup$ – Zach Teitler Aug 24 '18 at 16:41

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