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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12
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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 ...
6
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0answers
132 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 ...
6
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0answers
94 views

Proving the multiplicativity of a quaternary quadratic form

Consider the set $S$ of all integers of the form $f(x,y) + f(z,w)$, where $x,y,z,w$ are integers, $$ f(x,y) = a x^2 + b x y + a^2 y^2,$$ and $a,b$ are integers with $$a > 1, \; \; 0 < b < ...
5
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0answers
119 views

Non-degenerate bilinear forms of Lie algebra with a degenerate Killing form

Definition: A Lie algebra is defined by: $$ [e_a,e_b]={f_{ab}}^ce_c $$ The Killing form is $$ g_{ab}=-{f_{ac}}^d {f_{bd}}^c $$ Set-Up: The type of Lie algebra of our interests (found out during a ...
5
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0answers
85 views

Distribution for ratio of dependent quadratic forms.

Random vector $\mathbf{x}_{0}$ $\sim$ $\mathcal{N}\left(\boldsymbol{\mu}, \mathbf{\Sigma} \right)$ is a sum of two orthogonal random vectors: $\mathbf{x}_{0}$ = $\mathbf{x}_{1}$ + ...
5
votes
0answers
167 views

Solutions to equation in matrix form

Suppose $x=[x_1, x_2, \cdots, x_n]^t$, $b=[b_1,b_2,\cdots,b_n]^t$ with $b_i\in K$ and $A\in M_n(K)$, where $K$ is a field. There are well known criteria for the system of equations $Ax=b$, by ...
4
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0answers
154 views

Proving the max of a quadratic form ${\mathbf x}^T\mathbf A \mathbf x$ can be attained when $x$ is from $n$-dimensional hypercube

updated: Maybe my original question is somewhat misleading. I rewrite some of the post. This is some research problem I'm working on. I have an $n\times n$ symmetric positive-definite matrix ...
4
votes
0answers
272 views

Understanding a proof about Hilbert Matrix

EDIT: I asked 3 questions. The first one I was able to solve myself, and the other two I cross-posted to MO. Lately I've been interested in the Hilbert Matrix (its definition will come later). I went ...
4
votes
0answers
160 views

Proof that the Arf invariant is independent of choice of basis

I'm confused about the proof of the following claim: Let $V$ be a vector space of dimension $2n$ and let $e_i, f_i$ be a symplectic basis. Let $q: V \to Z_2$ be a non-degenerate quadratic form. ...
3
votes
0answers
34 views

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

Let $\alpha \in \mathbb{C}$ be an algebraic number. If the minimal plynomial of $\alpha$ over $\mathbb{Q}$ has degree $2$, we say $\alpha$ is a quadratic number. Then $\alpha$ is a root of a unique ...
3
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0answers
106 views

counterexample for: integral forms that are equivalent as rational forms are also equivalent as integral forms

I have the feeling that I'm missing something very obvious: I'm looking for a counterexmple for the following statement for some $n>1$ (it is trivially true for $n=1$): Let $A,B\in\mathbb ...
3
votes
0answers
563 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 ...
3
votes
0answers
66 views

Siegel's theorem

I want to learn the proof of the following theorem by Siegel. The statement of the theorem is taken from "Symmetric bilinear forms" by Milnor and Husemoller (pp. 44). They say that the proof is due to ...
2
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0answers
55 views

Solving a system of equation and finding the largest possible value of one of the variables

This problem comes from question 5 in the PUMAC Algebra A competition (link here): Suppose $w, x, y, z$ satisfy $$w+x+y+z=25$$ $$wx+wy+wz+xy+xz+yz=2y+2x+193$$ The largest possible value of $w$ can ...
2
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0answers
49 views

Connection between class number and the theory of Ideals/Quadratic Fields

I've been studying the classic results in integer binary quadratic forms, mainly the equivalence and reduction of quadratic forms and the class number $H(d)$ (the definition I got for $H(d)$ is the ...
2
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37 views

Definition of the term 'generic' in context of quadratic forms.

In Proposition 3.3 of the paper: A. Lubotzky, R. Phillips and P. Sarnak, Ramanujan Graphs, Combinatorica 8(1988), the authors use a result obtained by Malisev : "Let $f(x_1,\ldots,x_n)$ be a ...
2
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0answers
38 views

Classifing Second Degree Curves/Surfaces

I have got myself into a pickle with the following question: Classify the following (ellipse, hyperbola, ellipsoid etc) $x^2 + y^2 + 2z^2 + 2xz - 2y + 2z + 2 =0$ Now, I have written a symmetric ...
2
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0answers
33 views

Holzer reduction of solutions of quadratic ternary forms

Suppose $(x_{0}, y_{0}, z_{0})$ is a solution to the equation $ax^2 + by^2 + cz^2 = 0$. The solution is said to be Holzer reduced if $x_{0} < \sqrt{|bc|}$, $y_{0} < \sqrt{|ac|}$ and $z_{0} < ...
2
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0answers
32 views

$U$ is isotropic $\implies$ $U\subset U^0$

Let $(V,Q)$ be a quadratic module and $U$ is a subspace of $V$. Serre (A Course in Arithmetic, p. 29) claimed that the following is evident: $U$ is isotropic $\iff$ $U\subset U^0$. In other ...
2
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0answers
91 views

Is there a generalization of eigenvalues/eigenvectors to $L^p$ for $p\neq2$?

Given symmetric positive definite $m\times m$ matrix $\Sigma$, $1 \le p,$ and constant $c > 0$, consider the solution to the optimization problem: $$ \min_{v : \left\|v\right\|_p \ge c} ...
2
votes
0answers
152 views

A characterization of an ambiguous class of binary quadratic forms of discriminant $D$

We use the definitions of this question. Let $D$ be a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). There exists a bijection $\psi\colon Cl^+(R) \rightarrow C(D)$ by ...
2
votes
0answers
111 views

Quadratic transformations of vector spaces

Much is known about transformations of the following form $$y_i = L_{ij}x_j \;\;: \;\; x\in\mathcal{R}^n, L\in\mathcal{R}^{n\times n}$$ We can infer a number of geometric properties about the ...
2
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0answers
246 views

Surface Function Fitting to Spherical Data

I have a set of geographic (longitude,latitude,value) data to which I would like to fit surface functions, specifically, the set of quadratic surfaces: $f(x,y)=Ax^2+Bx^2+Cxy+Dx+Ey+F$ At the moment, ...
1
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0answers
58 views

Reducing a linear algebra expression to quadratic form

I am trying to solve the following exercise for my Machine Learning course. Expand this expression so that there are only quadratic terms: $(\mathbf{x} - \mathbf{\mu})^T \mathbf{\Sigma}^{-1} ...
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0answers
15 views

Distribution of a quadratic form

Let $A$ be a symmetric positive definite matrix, and $x$ a random vector. Suppose we know the distribution of $x^\top A x$. What can we say about the distribution of $x^\top x$?
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0answers
28 views

Classification of quadrtic forms over Q_p

I need some one to recap the topic with me and correct me when i am wrong. There are basically just a few questions at the end,but its important to also show what i know and not just what i dont. ...
1
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0answers
42 views

Problem about $\mathbb{P}^3(K)$

Show that four skew lines in $\mathbb{P}^3$ have two transversals in common. I know that exist a quadric which contains three of the four lines....but i'm stuck EDIT: If the skew lines are ...
1
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0answers
28 views

Approximation of a quadratic form

Let $\mathbf{x}=(x_1,\cdots,x_n)^T\in\mathbb{R}^n$ and $A\in\mathbb{S}_{++}^n$ be a symmetric positive definite matrix. Also, let $Q\colon\mathbb{R}^n\to\mathbb{R}$ be the quadratic form given by $$ ...
1
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0answers
15 views

Diagonalization of quadratic forms over $\mathbb{Q}$

I'm having difficulties in finding the diagonal forms of some quadratic forms. I am sure it is not supposed to be that difficult but I guess I am lacking some creativity after overdoze of coffee and ...
1
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0answers
58 views

Enumerating integer solutions to quadratic equations

Consider a quadratic equation with integer coefficients in two variables. $$ax^2+bxy+cy^2+dx+ey+f=0$$ I would like to know how to find the number of integer solutions $(x,y)$ to this equations. Is ...
1
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0answers
58 views

References for Composition Law on Binary Quadratic Forms

What are some good references for the Gauss composition law on binary quadratic forms in terms of Galois Cohomology? It is my understanding that there is a cohomological approach, and I am studying ...
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0answers
76 views

Does this quadratic form represent 1?

I am stuck on the following question in Lam's quadratic forms for a few days now. Let $a,b,c$ be three elements of a field $F$ such that $0 \neq a^2+b^2 \neq c^2$. Show that the quadratic form ...
1
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0answers
61 views

How to solve an optimization problem with non-convex Frobenius norm constraint?

The form of my problem is: $$ \min_W \|Y-WX\|_F^2-\|V-WU\|_F^2 $$ $$ s.t. \|W\|_F=1 $$ All five variables are matrices. Since the norm constraint is a non-convex one, I have no idea how to solve this ...
1
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0answers
16 views

proving there exist another basis of non-degenerate quadratic space (V,B) other than the given basis

If {$v_i$} is a basis of non-degenerate quadratic space ($V,B$) (finite), prove that there exists another basis {$w_i$} such that $$B(v_i,w_j)=1 (i=j)$$ $$or 0(i \neq j)$$ Sorry for the ugly text ...
1
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0answers
69 views

How to compute the class group of an order of a quadratic number field

Let $K$ be a quadratic number field. Let $R$ be an order of $K$, i.e. the subring of $K$ which is a free $\mathbb{Z}$-module of rank $2$.Let $D$ be its discriminant. We use the notation and the result ...
1
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0answers
96 views

Please help in solving $ax^2 + bxy + cx + dy + e$ = 0

Sometime back when trying to work out how to solve $ax^2 - by^2 + cx - dy + e = 0$ I learned that the way to solve such forms is to 'square the terms' and give it the form $A^2 - B^2 - E = 0$, $A = ax ...
1
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0answers
27 views

System of symmetric quadratic equations

Suppose $A_1, A_2, \ldots A_k$ are real symmetric (but possibly singular or indefinite) matrices. I want to know whether the system of quadratic equations $$v^T A_i v =0 $$ has a nontrivial solution ...
1
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0answers
80 views

About quadratic form and its discriminant

There are 3 parts of the problem. Let d be a perfect square, possibly 0. Show that there is a quadratic form $ax^2+bxy+cy^2=0$ of discriminant d for which a=0. Let a,b,c be integers with $a\ne0$. ...
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0answers
43 views

Witt index and Pfister neighbours

Suppose $\phi$ be a 8-dimensional quadratic form with trivial discriminant over a field $F$ of characteristic not 2. Assume that there is 3-fold Pfister form $<<a,b,c>>$ such that ...
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0answers
61 views

Convergence in distribution of a quadratic form

If $Q_n=X_nM_nX_n=\sum_{i,j=1}^n X_i m_{nij}X_j$, $X_n=(X_1,...,X_n)$ where $X_j$ are iid random variables and $M_n=(m_{nij})$ is a symmetric matrix with extending rownumber in $n\to\infty$. Iam ...
1
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0answers
30 views

Solve I.V.P for differential using quadratic form

Solve the i.v.p for $y''+4y'+5y=0, y(\frac{\pi}{2})=1/2, y'(\frac{\pi}{2})=-2$ I solved using the quadratic form. and I got $\lambda = \frac{(-4 \pm 2i)}{2}$, which for $\lambda 1,2= 2+2i$. And then ...
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0answers
24 views

Can we express a SPD matrix $S$ in terms of $S^{2}$ in a different manner to solve a convex problem?

I have to find the Symmetric Positive Definite matrix $S\in \mathcal{M}_{m,m}$ that minimizes the function $f(S)=\mathrm{trace}(S)+\mathrm{trace}(S^{-2})$ which has been proven to be convex in the ...
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0answers
108 views

The double summation in the general quadratic form

According to my book and Wikipedia, a quadratic form on $\mathbb{R}^k$ is a real-valued function of the form $Q(x_1,...,x_k)=\sum_{i,j=1}^{k} a_{ij}x_ix_j.$ When I try to use this to check the general ...
1
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0answers
233 views

quadratic form of trace_inverse of symmetric positive definite matrix

I have the following problem: I need to implement a program that doesn't accept the matrix quadratic form $B^T\times B$ but it accepts the scalar quadratic form instead. Actually I need to find a ...
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0answers
90 views

Minimize a complex quadratic subject to two convex quadratic constraints

I have the following the optimization problem \begin{align} \min_{\mathbb{x}\in \mathbb{C}^{N \times 1}}~&||\mathbb{x}^H\mathbb{u}||_2^2-2*Real\{ \mathbb{x^Hu}\} \\\ ...
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0answers
91 views

quadric surfaces. Uniqueness theorem

I would like to know references (book or monograph) for this theorem about quadric surfaces. If $F(x)=0$ is an equation in $K[x]$ of a quadric surface $C$ and i) ${\rm char}(K) \not= 2$ ii) $C$ has a ...
1
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0answers
132 views

solution count of quadratic form congruences

Let $(L,q)$ be an even Lorentzian lattice of signature $(1,l-1)$, i.e., $q(\lambda) \in \Bbb Z$ for all $\lambda \in L$ and the (non-degenerate) quadratic form $q$ is of type $(1,l-1)$ (the ...
1
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0answers
56 views

Witt Cancellation over $\mathbb{Z}/{p^e \mathbb{Z}}$?

I wonder whether someone knows if the Witt cancellation theorem also holds for the rings $\mathbb{Z}/{p^e \mathbb{Z}}$ where $p$ is an odd prime and $e \in \mathbb{N}$, i.e. for example, let $G = ...
0
votes
0answers
16 views

Fisher-Cochran's theorem

$dim(E)=n$ We have $u_1, u_2..., u_p$ self-adjoint operators which belong to $E$ $(i)$ : $rk(u_1)+...+rk(u_p)=n$ $(ii)$ : $q_1(x)+...q_p(x)=x.x$ with $q_i$ the quadratic form $q_i(x)=u_i(x).x$ for ...
0
votes
0answers
31 views

Show that for any given d<0, the primitive positive definite quadratic forms of discriminant d all have the same number of automorphs.

Show that for any given d<0, the primitive positive definite quadratic forms of discriminant d all have the same number of automorphs. I think we should let f(x,y) = $ax^{^{2}} + bxy + ...