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

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15
votes
5answers
733 views

Every integer vector in $\mathbb R^n$ with integer length is part of an orthogonal basis of $\mathbb R^n$

Suppose $\vec x$ is a (non-zero) vector with integer coordinates in $\mathbb R^n$ such that $\|\vec x\| \in \mathbb Z$. Is it true that there is an orthogonal basis of $\mathbb R^n$ containing $\vec ...
3
votes
2answers
316 views

Isotropy over $p$-adic numbers

Over what $p$-adic fields $\mathbb{Q}_p$ is the form $\langle3, 7, -15\rangle$ isotropic?
1
vote
3answers
851 views

How to solve inhomogeneous quadratic forms in integers?

If I have a quadratic form like $y^2 - x^2 - x = k$ none of the techniques I know work because of the nasty $x$. Note that homogenizing doesn't work because a solution of $Y^2 - X^2 - X Z = k Z^{(2)}$ ...
8
votes
0answers
171 views

What numbers are integrally represented by $4 x^2 + 2 x y + 7 y^2 - z^3$

EDIT: a selection of material relevant to this problem is available at INHOMOGENEOUS In December 2010 my question appeared in the M.A.A. Monthly, show that $4 x^2 + 2 x y + 7 y^2 - z^3 \neq \pm 2 ...
0
votes
2answers
158 views

How to solve an equation of the form $ax^2 - by^2 + cx - dy + e =0$?

I am trying to find out how to solve $ax^2 - by^2 + cx - dy + e = 0$ to get integer solutions, failing this the rational solutions. Thanks!
1
vote
1answer
83 views

Elementary properties of integral binary quadratic forms

Let $f = ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. $D = b^2 - 4ac$ is called the discriminant of $f$. We say $f$ is positive definite if $a \gt 0$ and $D \lt 0$(cf. this ...
7
votes
1answer
955 views

What's a BETTER way to see the Gauss's composition law for binary quadratic forms?

There is a group structure of binary quadratic forms of given discriminant $d$: Let $[f]=[(a,b,c)], [f']=[(a',b',c')],$ where $d=b^2-4 a c=b'^2-4 a' c'.$ The composition of two binary quadratic ...
26
votes
2answers
822 views

Elementary proof that $2x^2+xy+3y^2$ represents infinitely many primes?

We did in class $x^2+y^2$, which was easy, and we had for homework $2x^2+2xy+3y^2$, which I did (its values (if not square) must be divisible by form primes, or of the form $x^2+5y^2$, and clearly ...
12
votes
0answers
1k views

Proof of Legendre's theorem on the ternary quadratic form

Theorem (Legendre): Let $a,b,c$ coprime positive integers, then $ax^2 + by^2 = cz^2$ has a nontrivial solution in rationals $x,y,z$ iff ...
7
votes
2answers
1k views

Show $15x^{2} - 7y^{2} = 9$ has no integer solutions

I'm trying to show the quadratic binary has no integer solution. I've used the following process to transform it into a Pell's equation of the form $x^{2} - Dy^{2} = M$ If there is a solution, then ...
4
votes
2answers
140 views

Should isometries be linear?

Question Suppose $V$ is a (finite-dimensional) vector space over $F$ ($\operatorname{char }F\neq2$, due to user1551) equipped with a non-degenerate quadratic form $Q$, and $T$ is a ...
1
vote
5answers
271 views

Algorithm for determining whether two imaginary quadratic numbers are equivalent under a modular transformation

Let $\Gamma = SL_2(\mathbb{Z})$. Let $\mathcal{H} = \{z \in \mathbb{C}\ |\ \mathcal{Im}(z) > 0\}$ be the upper half complex plane. Let $\sigma = \left( \begin{array}{ccc} p & q \\ r & s ...
1
vote
7answers
368 views

positive definite quadratic form

Is $\sum_{i=1}^n x_i^2 + \sum_{1\leq i < j \leq n} x_{i}x_j$ positive definite? Approach: The matrix of this quadratic form can be derived to be the following $$M := \begin{pmatrix} 1 & ...
5
votes
3answers
381 views

Etymology of the word “isotropic”

Given a quadratic form $q : V \rightarrow k$, a nonzero vector $v \in V$ is said to be isotropic if $q(v) = 0$. Any subspace of $V$ containing such a vector is also said to be isotropic, and the ...
2
votes
1answer
149 views

A condition for an odd prime to be represented by a binary quadratic form of a given discriminant

Let $f = ax^2 + bxy + cy^2$ be an integral binary quadratic form. We say $D = b^2 - 4ac$ is the discriminant of $f$. If $D < 0$ and $a > 0$, we say $f$ is positive definite. It is easy to see ...
0
votes
3answers
145 views

Solving a system of nonlinear (quadratic) equations

Consider the following system of equations: $$\begin{align} (x + 1)^2 [(p - l)^2 + (q - m)^2] &= (a - l)^2 + (b - m)^2 \\ (x + 1)^2 [(p - a)^2 + (q - b)^2] &= x^2[(a - l)^2 + (b - ...
9
votes
2answers
510 views

Generalizing the 290 theorem.

I have only just come across the remarkable theorem of Conway about universal quadratic forms over $\mathbb{Z}$; namely that in determining whether a integer coefficient, positive definite quadratic ...
3
votes
0answers
595 views

Simultaneous diagonalization of quadratic forms

I would like to collect references (or direct quotations) about as many "simultaneous diagonalization" results in linear algebra as possible. Let $V$ be an $n$-dimentional ($n$ finite) vector space ...
1
vote
3answers
134 views

Solving $a^2+3b^2=c^2$

I'm looking for how to solve the equation $a^2+3b^2=c^2$ where $a,b,c$ are integers and $b$ is even, I'm looking for the algorithm used to solve this kind of equations, not just the solution. Regards ...
9
votes
4answers
383 views

Representing a number as a sum of at most $k$ squares

Fix an integer $k >0 $ and would like to know the maximum number of different ways that a number $n$ can be expressed as a sum of $k$ squares, i.e. the number of integer solutions to $$ n = x_1^2 + ...
8
votes
4answers
2k views

Determining if a quadratic is always positive

Is there a quick and systematic method to find out if a quadratic equation is always positive or may have positive and negative or always negative for all values to its variables? Say for a quadratic ...
6
votes
1answer
119 views

Expressing a quadratic form, $\mathbf{x}^TA\mathbf{x}$ in terms of $\lVert\mathbf{x}\rVert^2$, $A$

EDIT: This question is actually an attempt to solve this. Please take a look. Let $A$ be a symmetric postive-definite $n\times n$ matrix, i.e. $A\in\mathbb{S}_{++}^{n}$ Also, let ...
4
votes
2answers
102 views

Proving the multiplicativity of a binary quadratic form

Consider the set $S$ of all integers of the form $x^2+y^2+4xy$, where $x$ and $y$ are integers. How could one prove the set $S$ is closed under multiplication? I have tried the bashy brute force ...
3
votes
2answers
149 views

On the hessian matrix and relative minima

I'm asked to prove the following statement: Let $\mathbb{A} \subseteq \mathbb{R}^n$ an open subset and $f: \mathbb{A} \to \mathbb{R} / f \in \mathbb{C}^3 \wedge (P \in \mathbb{A} : \nabla f(P)=0)$. ...
3
votes
1answer
5k views

Quadratic equation -> matrix?

Problem: Find the EigenValues and EigenVectors of the matrix associated with quadratic forms $2x^2+6y^2+2z^2+8xz$. I know how to convert a set of polynomial equations to a matrix but I have no clue ...
5
votes
1answer
237 views

Question about $4$-manifolds and intersection forms

This is a question related to an earlier question of mine: I've been reading about topological invariants. Some of them are defined in terms of quadratic forms. My current understanding is: we can ...
4
votes
5answers
185 views

Looking for proof of no solution to 4-variable quadratic diophantine equation

Show there are no integers $a,b,c,d$ such that $$\begin{cases}1=9ac+3ad+3bc-16bd \\ 1=3ad+3bc+2bd \end{cases}$$ Motivation: The ideal $I=(3,1+\sqrt{-17})$ in $R=\mathbb{Z}[\sqrt{-17}]$ has the ...
4
votes
2answers
373 views

Clifford Algebras

What would be the best source to learn Clifford Algebras from? Anything online would suffice or any textual sources for that matter.. I'm interested in doing a project in the subject, but I'm not ...
4
votes
1answer
155 views

Centre of a quadric

I found the following sentence in my linear linear algebra book (affine and projective geometry): $Q:V \to \mathbb{K}$ is a quadric (quadratic function) and $\alpha\in Aff(V)$. $Aff(V)$ is the set of ...
4
votes
1answer
148 views

Help fixing my broken example of Arf invariant

I need help fixing a broken example I've come up with. In particular, I wanted to use the Arf invariant to distinguish two non-homeomorphic surfaces. That's the first part that's broken since there ...
4
votes
1answer
251 views

Map preserving indefinite scalar product must be linear

Let $V$ be a finite dimensional real vector space and $\langle\cdot,\cdot \rangle$ be a positive definite scalar product in $V$. It is well know that if a map $T:V \to V$ preserves ...
4
votes
1answer
464 views

When are two diagonal matrices congruent?

This is probably a question that does not admit a simple answer. However, I'd like to know whether there exist criteria that determine when two diagonal matrices are congruent. I have the suspicion ...
2
votes
2answers
100 views

Is it true that the whole space is the direct sum of a subspace and its orthogonal space?

Problem The ground field is $K$, $\operatorname{char}K\neq2$. Suppose $W$ is a (maybe infinite dimensional) subspace of a vector space $V$ with a symmetric/symplectic form ...
1
vote
1answer
32 views

Discriminant of a Quadratic form

Let $V$ be a vector space over field $K$ and $Q$ is the quadratic form on it, and $A$ be the matrix w.r.t. $e_1,e_2,...e_n$ of $V$. Now $discr(Q)$ is defined as $det(A)$ mod ${K^{*}}^{2}$. Now my ...
1
vote
1answer
168 views

Let $W \subset V$ vector spaces, let $W'$ be the orthogonal complement of $W$. Show $\dim(W)+\dim(W')=\dim V$

As title says: Let $W \subset V$ vector spaces, let $W'$ be the orthogonal complement of $W$. Show $\dim(W)+\dim(W')=\dim V$. We are given that $W \subset V$ finite vector spaces, symmetric bilinear ...
1
vote
2answers
91 views

independent chi squares mean independent non central chi square?

Let $Y$ be a multivariate normal random vector with covariance $\Sigma$. Let $A_0,A_1$ be matrices such that $$A_0\Sigma A_1=0.$$ It is known that in this case $Y'A_0Y$ and $Y'A_1Y$ are independent ...
1
vote
3answers
84 views

Sum of two quadratic forms

Suppose I have two quadratic forms $Q_i(x)=(x-a_i)^T A_i(x-a_i)+c_i$, $i=1,2$ where $x,a_i \in \Bbb{R}^n$ and $A_i$ are positive-definite $n\times n$ matrices. Then $Q(x)=Q_1(x)+Q_2(x)$ is also a ...
1
vote
2answers
227 views

Isometry without injection and surjection

Suppose that $B_1$ and $B_2$ are bilinear form on space $V_1$ and $V_2$. An isometry relative to $B_1$ and $B_2$ is an linear map $\sigma:V_1 \rightarrow V_2$ satisfying ...
1
vote
1answer
153 views

Find an optimal 4-tuple which satisfies a boolean expression

(I have post a question with bounty for several days (Find a best 4-tuple which fulfils a variable boolean formula), but unfortunately got no answer yet. Here I simplify it to a smaller problem and ...
0
votes
1answer
185 views

Ratio of dependent chi squared random variables

Suppose that $X=v'A_1v$ and $Y=v'A_2v$, where $A_i$ are symmetric matrices and $v$ a multivariate normal vector with covariance $V$, are chi squared distributed each with its own degrees of freedom. ...
0
votes
1answer
463 views

Canonical form of a Matrix

My understanding of canonical form is very limited, and so may require some help. Suppose a quadratic of the form: $$ x_1*x_2+x_1*x_3=Q.$$ How would one go about putting that into canonical form, ...
-1
votes
2answers
470 views

Conditions for a real binary quadratic form to be positive definite

Since this question was heavily downvoted, I would like to change the presentation of the question as follows. I hope those of you who downvoted this question would be satisfied with the change. In ...
-1
votes
1answer
144 views

Relation between an integer represented by a binary quadratic form and a certain Dirichlet character defined by Jacobi symbol

Let $f = ax^2 + bxy + cy^2$ be a binary quadratic form over $\mathbb{Z}$. We say $D = b^2 - 4ac$ is the discriminant of $f$. It's easy to see that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). If ...
-1
votes
1answer
65 views

Standardize a Quadratic Form

standardize Quadratic Form $$8x_1x_4+2x_3x_4+2x_2x_3+8x_2x_4$$ how to do? what's the simplest method. How to choose the first linear replacement, matrix or do ...