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 ...
9
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195 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
78 views

Official name of Fermat's $x^2+3y^2$ theorem?

One of Fermat's more well-known claims is that for any prime number $p$, $p=x^2+3y^2\iff p\equiv 1\pmod 3$. Does this have an "official" name? (Another one which goes $p=x^2+y^2\iff p\equiv 1\pmod 4$ ...
6
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0answers
137 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 ...
6
votes
0answers
102 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
votes
0answers
96 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
169 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
votes
0answers
31 views

Quadratic optimisation with quadratic equality constraints

I would like to solve the following optimisation problem: $\min_{x} (x'Ax)$ subject to $x'Bx = x'Cx = 1$. Where A is symmetric and B and C are diagonal. Does anyone have a suggestion for an ...
4
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0answers
208 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
117 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 ...
4
votes
0answers
346 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
824 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 ...
4
votes
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168 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
36 views

(Fast) eigen decomposition of $DXD$ where $D$ is diagonal, $X$ is symmetric with known eigen decomposition

Assuming that I already know the eigen-decomposition of a real symmetric matrix $X$, is there any way to use it to retrieve efficiently the eigen-decomposition of $DXD$, where $D$ is a diagonal ...
3
votes
0answers
68 views

quadratic form from nxn matrices to reals ( Tr(A^2) ). I need to find it's signature and rank.

Firstly prove $Tr(A^2)$ defines a quadratic form from the space of $n \times n$ matrices to R. I think you just have to show that $Tr(A B)$ is a bilinear form which seems too easy to be correct or I'm ...
3
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0answers
82 views

A Tale of Two Quadratic Identities (Pell-like)

Question is at the end. Let all variables be integers. For some constants $a,b,c,d$, assume we have initial solution {$m,n$} to, $$a m^2 + b m n + c n^2 = d\tag{1}$$ Identity 1: $$a x^2 + b x y + ...
3
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0answers
52 views

Freeman Dyson's identity for the modular discriminant $\Delta$

In his 'Introduction to Modular Forms', Don Zagier states the Freeman Dyson's identity : $$\Delta(\tau)=\sum_{\substack{ (x_1,\ldots,x_5)\in \mathbb{Z}^5 \\ x_1+\cdots+x_5=0 \\ x_i \equiv i ...
3
votes
0answers
36 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
votes
0answers
234 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 ...
3
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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
49 views

Trying to prove a theorem on simultaneous diagonalisation of matrices

Let $B$ be a $n\times n$ real symmetric positive definite form, and $A$ be a $n\times n$ real symmetric form. There exists an orthogonal matrix O such that $O^TBO=I$ and ...
2
votes
0answers
15 views

Witt Groethendieck Ring splitting

I have a really basic question about the Witt Groethendieck ring of a field: In Lam's book, it says that $\hat{W}(F)/\hat{I}^2(F)$ depends only on the square classes of $F$, $\hat{W}/\hat{I}^2\cong ...
2
votes
0answers
45 views

Alternative way to recognize that a real symmetric quadratic form is positive

A real symmetric quadratic form $g(x)=\varSigma_{1\leq i,j \leq n}\,g_{ij} x^i x^j$ is positive (definite) if $g(x)>0$ for every $\mathbb{R}^n\ni x=(x^1,...,x^n)\neq 0$. It is well known that a ...
2
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0answers
31 views

Objects corresponding to Higher forms

If $Q$ is a quadratic form, then we know there exists matrix $A$ such that $Q=xAx'$ and $Q$ can be expressed as weighted sum of eigenvalues of $A$. If $H$ is a higher order form, then is there an ...
2
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0answers
30 views

can someone break this quad formula down for me?

Can someone explain how this person yield the stuff on the right side using quad formula?
2
votes
0answers
65 views

Law of large numbers for linear (quadratic) combinations of i.i.d. random variables

Let $(X_i)_{i\in\mathbb{N}}$ be i.i.d. real random variables with zero mean. By the law of large numbers $$\frac{1}{n}\sum_{i=1}^nX_i \to 0 \quad\text{(almost surely, in probabability...) as ...
2
votes
0answers
55 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
votes
0answers
48 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
votes
0answers
59 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
votes
0answers
37 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
35 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
votes
0answers
94 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
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0answers
128 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
votes
0answers
270 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
vote
0answers
46 views

Computing the matrix representation of the quadratic form $A \mapsto \text{tr}(A^2)$

Define the quadratic form $Q:\mathbb{R}^{2\times 2}\to\mathbb{R}$ by $$Q(A) = \text{tr}(A^2).$$ What is the matrix representation of this bilinear form with respect to the standard basis of ...
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0answers
20 views

Maximizing ratios of quadratic forms with several norms

to maximize a ratio of quadratic forms, $(u^\top Mu) / (u^\top Ku)$, or a canonical correlation analysis ratio $(u^\top Rv) / [(u^\top Ku)^{1/2} \times (v^\top Lv)^{1/2}]$, is done straightforwardly ...
1
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0answers
49 views

Pell's equation and binary hyperbolic forms.

We define the function $f:\mathbb Z_p \times \mathbb Z_p \rightarrow \mathbb Z_p$ such that $f(x,y)=x^2-cy^2$ and $0\neq c=a^2+b^2$. Is it true that $f$ is hyperbolic? In other word's is there any ...
1
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0answers
28 views

An inequality from $0-1$ matrices

Let $A\in\{0,1\}^{n\times n}$ of real rank $r$. Let $J$ be all one matrix. Denote $\underline{x}=(x_1,\dots,x_n)$, $\underline{y}=(y_1,\dots,y_n)$. It is clear we have ...
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0answers
26 views

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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0answers
21 views

Reference about quadratic forms with discriminant 1

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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0answers
28 views

Question on positive definiteness of non homogeneous quadratic form

I'm having trouble understanding a proposition from a semidefinite-programming textbook. It goes as follows: Let $Q$ be a quadratic function of $x \in \mathbb{R}^n$ given by $$Q(x) = ...
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0answers
35 views

Albert- Algebras and Traceforms

Im new to the topic so this could be basic nonsense to you. Any Albert-Algebra $A$ has a trace map $T:A \rightarrow k$ and thus one can assign a quadratic form $q_A$ of rank $27$ by setting $q_A(x) = ...
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0answers
47 views

Show Equivalence of Binary Quadratic Forms

I've been stuck on these two problems from my problem set for quite a while. Any help would be appreciated! 2)Suppose that $ax^2 + bxy + xy^2$ is equivalent to $Ax^2 + Bxy + Cy^2$. Show that $gcd ...
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0answers
42 views

Theorem for Equal Sums of Like Powers $x_1^8+x_2^8+x_3^8+\dots$

Kindly see the question at the end of post. Solutions to the system of three equations, $$\begin{aligned} a^2+b^2+c^2+d^2\, &= e^2+f^2+g^2+h^2\\ a^4+b^4+c^4+d^4\, &= e^4+f^4+g^4+h^4\\ abcd\, ...
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0answers
30 views

Reduce degree of a high degree unconstrained binary term to quadratic unconstrained binary term

I'm working on a optimization project, in this project I have to convert higher order unconstrained binary polynomial to quadratic unconstrained binary polynomial. Can anyone give me a hint of how to ...
1
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0answers
118 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} ...
1
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0answers
18 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
75 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 ...
1
vote
0answers
37 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
vote
0answers
45 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 ...