# Tagged Questions

Questions about quadratic forms in multiple variables, for example $4x_1^2 + 3x_1x_2 + 5x_1x_3 - 8x_3^2$.

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### A problem on unitary spaces

If $V$ is a unitary space with a hermitian form $\langle,\rangle$ and $v_1,...v_n$ are any $n$ vectors in $V$ then is it true that ${\rm det}(\langle v_i,v_j\rangle)\geq 0$? When does equality hold?
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### Finding integers of the form $3x^2 + xy - 5y^2$ where $x$ and $y$ are integers, using diagram via arithmetic progression

So the diagram drawn looks like this: We begin at the edges labeled $3$ and $-5$ because we are using those as the bases for $x$ and $y$, respectively. The way we obtain the values of the 2 ...
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### Developping a long quadratic form like $(x-y-z-\mu)^t\Sigma^{-1}(x-y-z-\mu)$

Is there a way to some how develop a long quadratic form ? Maybe something like : $(x-y-z-\mu)^t\Sigma^{-1}(x-y-z-\mu)= (x-\mu)^t\Sigma^{-1}(x-\mu) - (y+z)^t\Sigma^{-1}(y+z)$ or is there another way ...
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### What's so special about the form $ax^2+2bxy+cy^2$?

Binary quadratic forms are sometimes studied (e.g. by Gauss) in the form $$ax^2+2bx+cy^2$$ In other words, the second coefficient is assumed to be even, and the polynomial is assumed to be ...
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When I am reading Serre's $A$ $Course$ $In$ $Arithmetic$, Chapter 5, it deals with $quadratic$ $forms$ of some vector space $V$, which can be viewed as an extension of an $abelian$ $group$ $E$ of ...
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Two quadratic forms $$Q(x_1, x_2, \dots , x_n) \\ \text{ and } Q'(x_1, x_2, \dots , x_n)$$ are called equivalent $$\Leftrightarrow Q'(x)=Q(Tx), \text{ where } T \in M_n(K), \text{ invertible }$$ ...
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### If I have a polynomial $x^2(1-m^2) - x2m^2 - (m^2 + 1)$ with a solution at $x = -1$, how do I get the other root

If I have a polynomial $x^2(1-m^2) - x2m^2 - (m^2 + 1)$ with a solution at $x = -1$, then I know I can just take $x^2(1-m^2) - x2m^2 - (m^2 + 1)$ and divide it by $x+1$ to get the other root. In a ...