# Questions tagged [quadratic-forms]

A quadratic form is a homogeneous polynomial of degree two (in any number of variables), for example $4x_1^2 + 3x_1x_2 + 5x_1x_3 - 8x_3^2$.

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### Assumption $d>2$ on Proposition 2.12 from Knapp's Elliptic Curves

I'm going through Knapp's book on elliptic curves and I got stuck in a minor detail. This is a part of the proof of Proposition 2.12: I could understand everything except for this little detail: ...
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### Find parameter a for which..

Find parameter $a$ for which $$\frac{ax^2+3x-4}{a+3x-4x^2}$$ takes all real values for $x \in \mathbb{R}$ I have equated the function to a real value, say, k which gets me a quadratic in x. I have ...
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### Finding representative matrix of quadratic form $q(x) = x_1x_2 + x_2x_3 + x_1x_3$ with respect to a basis

I'm slightly confused about this question: we have a basis of $R^3$ $B =${$e_1,e_2, e_3$}, where $x = x_1e_1 + x_2e_2 + x_3e_3$. We need to give the representative matrix of $q$ with respect to ...
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### Positive-definite over $\mathbb Q$ form is positive definite over $\mathbb R$?

I was reading P. Etingof's "Introduction to the representation theory" when I found this problem (and I've trouble with it): we have a quadratic form $Q(x) = \sum x_i^2 - \frac{1}{2}\sum b_{ij}x_ix_j$,...
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### Signature of a bilinear form in the space of $2 \times 2$ Hermitian matrices

We have a bilinear form, $$\langle A, A' \rangle = \det(A+A') - \det A - \det A'$$ on the real vector space of Hermitian $2\times 2$ matrices. If I have a basis of the space, I would calculate ...
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### Make Ising-like quadratic form positive definite

Consider a quadratic form $Q_a:\mathbb{C}^N\rightarrow\mathbb{C}:s\mapsto Q_a(s)=(a+it)\sum_{i=1}^N s_i^2-\beta J\sum_{\langle i,j\rangle}s_is_j$. The idea here is that at every point $i$ of a lattice ...
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### Form Class group - Special cases

I am trying to find special cases for when the form class group will have a predictable structure. I am specifically interested in the case of prime discriminants or relating the structure for non-...
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### Quadratic Forms Orthogonal Diagonalization Existence

Why does one assume that the eigenbasis for a quadratic form is orthogonal, hence orthogonal diagonalization. I understand that for hermitian and unitary maps one can show by spectral theorem an ...
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### Deterministic algorithm for class group of Binary quadratic forms

I want to find the class group of a given negative discriminant. I know of Shanks method but this is not deterministic. I can of course brute force the problem by finding the class group and then ...