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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23 views

Quadratic form with one value changed.

I came across the following problem when trying to run a Metropolis algorithm. (It is related to computing a multivariate normal density.) Let us have an $n\times n$ matrix $A$ of a special kind: ...
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1answer
39 views

Quadratic Forms Using Derivatives

This link says we can diagonalize a quadratic form $$ f(\vec{x}) = \sum_{i,j=1}^n a_{ij}x_i x_j, $$ $$a_{ij} = a_{ji}, a_{ii} \neq 0$$ using derivatives (?!!!) in a formula like $$f(\vec{x}) = \...
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2answers
95 views

Surface described by the equation $-3y^2 - 4xy + 2xz + 4yz - 2x - 2z + 1 = 0$

Given the equation : $-3y^2 - 4xy + 2xz + 4yz - 2x - 2z + 1 = 0$. Check if the surface described by that equation has a center of symmetry and then by making the correct coordinate system change, find ...
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1answer
43 views

Geometric meaning of Equation

As a part of my linear-algebra exam preparation, I am going through the surface equation and quadratic-bilinear form usage in my book which is a part we haven't really went through and left to explore ...
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4answers
153 views

Solutions to a quadratic diophantine equation $x^2 + xy + y^2 = 3r^2$.

Let $k,i,r \in\Bbb Z$, $r$ constant. How to compute the number of solutions to $3(k^2+ki+i^2)=r^2$, perhaps by generating all of them?
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1answer
2k 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 $\left(\frac{-bc}{a}\right)=\left(\frac{-ac}{b}\right)=\left(\...
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2answers
40 views

Reference request: Binary quadratic forms

I am currently a first year grad student doing an independent study on topics in algebraic number theory and am currently looking at some of the properties of the polynomial $n^2 + n + A$, where $A \...
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3answers
46 views

Finding the matrix of this particular quadratic form

I have been working on problems related to bilinear and quadratic forms, and I came across an introductory problem that I have been having issues with. Take $$Q(x) = x_1^2 + 2x_1x_2 - 3x_1x_3 - 9x_2^...
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0answers
15 views

Positive definite quadratic form defines nondegenerate positive inner product

The background to this question (which is not important for the actual question!) is that I'm working on something in geometric dynamics, specifically the Jacobi metric. To me it seems that one uses ...
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1answer
24 views

Extending a $q$-isometry

Let $U,W$ be maximal completely isotropic subspaces of a finitely dimensional quadratic space $(V, q)$ over a field $\mathbb{K}$, $\operatorname{char} \mathbb{K} \neq 2$. Prove that any $q$-isometry $...
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2answers
46 views

Describe an equation geometrically

Finishing the last few stuff left for my end-term semester exams on Linear Algebra II, I bumped across a collection of identical exercises, posting one below : Describe geometrically, giving as much ...
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0answers
38 views

Reduction of quadratic form to canonical form with Lagrange method

I have to do what title says, but also find a change of basis matrix from standard basis to canonical basis. $Q(x_1,x_2,x_3) = x_1^2 + 2x_1x_2 + 2x_1x_3 + 2x_2x_3 =$ $=(x_1+x_2+x_3)^2 - x_2^2 - x_3^...
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0answers
28 views

Number of singular lines for the quadratic form

Suppose $V$ is a $2n$-dimensional vector space over $\mathbb{F}_q$ with a quadratic form $Q$ of hyperbolic type defined on it. That is, if $x = (x_1, ..., x_{2n}) \in V$, then $$Q(x) = x_1 x_2 + ... +...
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1answer
69 views

Is there a Fermat-era proof of Theorem 69 from Dickson's Intro to NT?

In Dickson’s Introduction to the Theory of Numbers (Ch. VI, pp. 91-93), he gives the following [wonderful and wonderfully general] theorem. Theorem 69: All integral solutions of $$x^2-my^2=zw$$ ...
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0answers
24 views

Show $|E_1(\mathbb{F_q})|+|E_2(\mathbb{F_q})|=2(q+1)$

...under the assumption that $E_1,E_2$ are elliptic curves over $\mathbb{F_q}$ and that there is a (surjective) isogeny $\pi:E_1\rightarrow E_2$ defined over $\mathbb{F_{q^2}}$ obeying $\pi\phi_1=-\...
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1answer
28 views

Solutions of quadratic equation with n variables

I'm trying to find the roots of a quadratic equation with $n$ variables. I've looked through the internet but I wasn't able to find any convincing formula. Given a vector $v=${$x_1, x_2, x_3, ..., ...
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1answer
26 views

Prove or disprove: quadratic form $Q(v):=\beta(v,v) $ (biliniarform) is nondegenerate if $\beta$ is.

Let char$K \not = 2$ and $\beta$ be a bilinearform (not necessarily symmetric) on a $K$-vectorspace $V$. Let $Q$ be defined by ($v, w \in V$ arbitrary)$$Q(v) = \beta (v,v)$$ I've shown that $Q$ is a ...
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3answers
680 views

Positive symmetric matrices and positive-definiteness

Is a symmetric real matrix with diagonal entries strictly greater than $1$ and off-diagonal entries positive but strictly less than $1$ necessarily positive-semidefinite?
3
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1answer
55 views

show quadratic forms $x^2 + y^2 + z^2$ and $ x^2 - y^2 - z^2$ are equivalent over finite fields $\mathbb{F}_p$

Can I show the diagonal matrix (1,1,1) and (1,-1,-1) are equivalent over the finite field $\mathbb{F}_3$ Can I show the quadratic forms $x^2 + y^2 + z^2$ and $x^2 - y^2 - z^2$ are equivalent over the ...
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1answer
34 views

(Geometric) Intuition behind Different Types of Rank 2 Tensor (Specifically Quadratic Forms)

This is essentially a follow-up to this question: Differences between a matrix and a tensor I think I have a good intuition/idea for the change of basis for a rank-(1,1) tensor ($A\vec{v} = \vec{w}$) ...
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1answer
28 views

A specific case of quadratic forms

I have a quadric as follows: $$ax^2+by^2+bz^2+yz=0.$$ I am curious to know which shapes in $\mathbb{R}^3$ this equation describes for different value of $a$ and $b$?
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2answers
56 views

Lower bound on quadratic form

Suppose I have a non-symmetric matrix $A$ and I can prove that $x^T A x = x^T \left(\frac{A+A^T}{2}\right) x>0$ for any $x \ne 0$? Can I then say that $x^T A x \ge \lambda_{\text{min}}(A) \|x\|^2 &...
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2answers
28 views

How do I convert a quadratic form to a diagonal form?

I don't understand how I should choose the transformations to convert a quadratic form to a diagonal form. Ex: $x_1\cdot x_2 + x_1\cdot x_3 + x_2\cdot x_3$
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1answer
88 views

How to deduce the formula for quadratic form?

I almost every book about quadratic form we can see it described as following function: $$ f(x) = \frac{1}{2}x^T A x - b^Tx + c $$ My question is: How can we deduce this formula? I understand, ...
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0answers
22 views

Quaternary quadratic modular problem.

Consider quadratic form $$Q(w,x,y,z)=w^2-x^2-y^2+z^2$$ and fix $r\in(0,\frac12)$ and pick a large enough $n\in\Bbb N$. How do we find a solution to $$Q(w,x,y,z)\bmod n=0$$ on condition that $$\sqrt n\...
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0answers
32 views

QR decomposition for nondegenerate quadratic form

Let $A$ be an invertible real $n\times n$-matrix, and $q$ be a nondegenerate quadratic form on $\mathbb{R}^n$. Do we have the QR decomposition for $q$ ? In other words : is it true that there exists ...
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5answers
999 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 $\vec x$...
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1answer
20 views

Find an orthogonal Matrix to a quadric

Given the following Quadric $$F_4 := \{X \in R^3 | x_1x_2+x_1x_3+x_2x_3 =4\} $$ My task is find an orthogonal Matrix C and $d_1,d_2,d_3 \in R $ so that $$F_4 = C*\{Y \in R^3 | d_1y_1^2 +d_2y_2^2+...
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2answers
54 views

Check equivalence of quadratic forms over finite fields

How to check whether the two quadratic forms \begin{equation} x_1^2 + x_2^2 \quad \text{(I)}\end{equation} and \begin{equation} 2x_1x_2 \quad \text{(II)} \end{equation} are equivalent on each of ...
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0answers
13 views

Indefinite Boolean Quadratic Programming: number of minima

The Boolean Quadratic Programming problem is defined as: $\min_{x} f(x) = x^TQx + c^Tx$ s.t. $ x \in \{0,1\}^n$ It is a well-studied NP-Hard problem with many approximation algorithms proposed. I ...
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1answer
47 views

Solution to a simple system of quadratic equations

I am hoping to find a closed-form solution to the following system of $n$ quadratic equations: $$ x_j^2 = \sum_{i=1}^n B_{ij}x_i $$ for $j\in\{1,\dots,n\}$, where $B_{ij}\geq 0$. There is a trivial ...
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0answers
25 views

How to convert an equation to quadratic format

How can I convert the following equation into quadratic format, e.g., $x^TQx$: $\sum_{i=1}^n (\delta_i .x)^TA (\delta_i .x)$, where $\delta_i$ is indicator function of size $T \times 1$, same size ...
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0answers
14 views

$rad_{R}(V) = {0}$ if and only if $rad_{L}(V) = {0}$

How to prove that $rad_{R}(V) = {0}$ if and only if $rad_{L}(V) = {0}$? for V is a finite dimensional vector space? for V is infinite dimensional vector space?
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1answer
43 views

Degenerate quadratic form [closed]

I'm beginning with quadratics forms and I am wondering : Let $a$ be a real number and $q:\mathbb{R^4} \rightarrow \mathbb{R}$ given by $$(x,y,z,t) \rightarrow ax^2+2axy+y^2+4zt-at^2.$$ I would ...
3
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1answer
75 views

Pell's equation and representation elements of $\mathbb Z_p$.

We defined the function $f:\mathbb Z_p \times \mathbb Z_p \rightarrow \mathbb Z_p$ such that $f(x,y)=x^2-cy^2$ and $c\not\equiv 0\pmod{p}$. Is it true that $f$ is onto?
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1answer
30 views

Vector space $V$ , quadratic form $f :V\to R$ . Excercise on rad(F) and a new function.

Let $V$ be a finite vector space and $f:V\to R$ a quadratic form. $F$ is the linear symmetrical form of the quadratic $f$. a) Show that the subset $W = \{ w \in V \mid F(w,v) = 0 \text{ for every } v ...
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1answer
17 views

Bounds on a quadratic form

I am currently in the middle of a proof where it would be nice to have some estimates on the size of a quadratic form. In particular, I am looking at $$x^TAx$$ where $A$ is "small" (in the analyst's ...
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0answers
31 views

Completing the Square of Quadratic Forms

I was working through a proof of a lemma that lets us determine whether a Hessian is positive definite for Mardens' Vector Calculus, page 175 Basically the lemma is if $B= \begin{bmatrix} a &...
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3answers
128 views

Integer solutions to $x^2-xy+y^2=1$

What are the integer solutions to $x^2-xy+y^2=1$? (I found the solution below while working on another problem, so I thought I'll add it to the knowledge base here.)
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58 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, is there any ...
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0answers
16 views

Quadratic forms with the same discriminant

Show that any quadratic form $ax^2+bxy+cy^2$ with discriminant $-4$ is equivalent to the form $x^2+y^2$, and any quadratic form of discriminant $-3$ is equivalent to $x^2+xy+y^2$. Here quadratic ...
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0answers
15 views

Approximating a quadratic term in the constraint set as 2nd order Taylor expansion

I have an optimization problem in the following form: $$\min_{x,y} f(x)+g(y)$$ $$s.t.$$ $$Ax+h(y)=0$$ where $h(y)$ is a quadratic in $y$. Instead of solving this problem directly, $h(y)$ is ...
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2answers
2k views

diagonalize quadratic form

I have this quadratic form $Q= x^2 + 4y^2 + 9z^2 + 4xy + 6xz+ 12yz$ And they ask me: for which values of $x,y$ and $z$ is $Q=0$? and I have to diagonalize also the quadratic form. I calculated ...
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0answers
27 views

Sesquilinear Forms

I was trying to solve some exercises related to sesquilinear forms: Let V be a C-vector space (C - complex numbers) Prove that the set $\mathcal{S}(V)$ of sesquilinear forms on V is a vector ...
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0answers
21 views

Maximization of quadratic form on a sphere [duplicate]

I have to following problem $$\max_{x}x^TAx+b^Tx\quad \mathrm{s.t.}\quad x^Tx\leq c,$$ where $A$ is real, symmetric and positive semi-definite. Firstly I tried to solve the problem with the KKT, but ...
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2answers
197 views

Calculus approach to solve this Quadratic equation problem

Both roots of the equation $$(x-b) (x-c) +(x-a) (x-c) +(x-a) (x-b) = 0$$ are always positive , negative or real. Prove your result. By solving this equation I got $3x^2 - 2(a+b+c)x +ab + bc + ca = ...
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1answer
40 views

Show that if a quadratic form is primitive then so are equivalent forms

A Quadratic form is primitive if the greatest common divisor of the coefficients of it's terms is 1. I saw in number theory book that "it is easily seen that any form equivalent to a primitive form ...
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0answers
26 views

what is a ordinally quadratic function?

A function is ordinal equivalent to another means there exist a (unique) monotonic transformation between wiki definition of ordinal utility. I am a little confused, a function is ordinally quadratic ...
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0answers
10 views

Maximize function symbolically

I have the following expression: $$ \sum_{i,j=1}^n\rho_{ij}^2-\frac{2}{n}\sum_{i=1}^n\left(\sum_{j=1}^n\rho_{ij}\right)^2 +\frac{1}{n^2}\left(\sum_{i,j=1}^n\rho_{ij}\right)^2 $$ My goal is to ...
7
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2answers
114 views

Solving a quadratic 9-equation system

I need to solve the following system: $$\begin{cases} A^TA=B &(1)\\ A\vec{x}=\vec{y} &(2)\\ \end{cases} $$ I need $A$, given $B$, $\vec{x}$ and $\vec{y}$. $A$ and $B$ are both 3-by-3 ...