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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2answers
37 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 ...
0
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2answers
27 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: $x_1\cdot x_2 + x_1\cdot x_3 + x_2\cdot x_3$
0
votes
1answer
66 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, ...
2
votes
0answers
35 views

show diagonal matrix (1,1,1) and (1,-1,-1) are equivalent over $\mathbb{F}_3$

Can I show the diagonal matrix (1,1,1) and (1,-1,-1) are equivalent over the finite field $\mathbb{F}_3$ One way to phrase this is that $2\times 2$ matrices and Hamilton quaternions over ...
0
votes
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 ...
0
votes
0answers
24 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 ...
18
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5answers
992 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 ...
0
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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 ...
1
vote
2answers
49 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 ...
0
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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 ...
3
votes
1answer
41 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 ...
13
votes
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 ...
0
votes
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 ...
0
votes
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?
1
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1answer
40 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
votes
1answer
74 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?
0
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1answer
28 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 ...
0
votes
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
30 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 ...
2
votes
3answers
125 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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0answers
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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
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 ...
0
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0answers
22 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 ...
0
votes
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 ...
2
votes
2answers
195 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
39 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 ...
0
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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
votes
2answers
113 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 ...
0
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2answers
44 views

Representations of some primes as $3x^2-4y^2$?

I am looking for (elementary) proofs (idea of the proofs is also OK) or references to proofs of the followings: $$ p\equiv11\pmod{12}\longrightarrow p=3x^2-4y^2 $$ Any help appreciated.
0
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0answers
18 views

Completing a multivariate square

A well-known trick when analyzing quadratic polynomials $P=ax^2+bx+c$ is to complete the square: P can be written as $$P=\left(x+\frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a^2}.$$ I have been trying to ...
1
vote
1answer
71 views

Transforming Diophantine quadratic equation to Pell's equation

I have been discussing the fastest and most efficient ways of solving QDEs in a separate question record (Alternative method to solve quadratic Diophantine equations). However, as suggested by ...
0
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1answer
14 views

Factoring binary quadratic form in two second order polynomials

I have a binary quadratic form in $N$ and $D$, $AD^2 + BND + CN^2$, where $A$, $B$, and $C$ are real coefficients and $N$ is a second order polynomial of $x$ with real roots $\lvert r \rvert <1$ ...
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votes
1answer
20 views

Quadratic forms matrices

Let $$Q(x,y,z) = – 2x^2 + 6xy + 8y^2 + z^2.$$ Find the symmetric matrix associated with this quadratic form. Use the determinant method to determine whether the quadratic form is positive definite, ...
2
votes
0answers
54 views

On the integer solutions to $u^2+163v^2=w^3$ and others

It seems the solution of, $$u^2+dv^2 = w^3\tag1$$ involves the class number $h(d)$. Assume $\gcd(u,v)=1$. Q: For which $\color{red}{prime}\; d$ is the complete solution of $(1)$ in the integers ...
6
votes
0answers
38 views

Why do isotropic spaces deserve their name?

Wiki defines a quadratic form to be isotropic if it evaluates to zero at some vector. What does this have to do with isotropy in physics i.e uniformity in all directions? From my experience so far, ...
1
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1answer
62 views

Maximization of vector norm under a quadratic convex inequality constraint

I need help for the following problem: $$ \max_x x^Tx\quad \mathrm{s.t.}\quad x^TAx+b^Tx\leq c, $$ where A is symmetric, square and positive semidefinite, c is a real scalar and b is a real vector. ...
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2answers
100 views

Unable to find solution for $a^2+b^2-ab$, given $a^2+b^2-ab$ is a prime number of form $3x+1$

I have a list of prime numbers which can be expressed in the form of $3x+1$. One such prime of form $3x+1$ satisfies the expression: $a^2+b^2-ab$. Now I am having list of prime numbers of form $3x+1$ ...
7
votes
3answers
66 views

Fermat's Challenge of composition of numbers

In his letter to Carcavi (August 1659), Fermat mentions the following challenge There is no number, one less than a multiple of $3$, composed of a square and the triple of another square. ...
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1answer
58 views

Regarding the factorization $a^2+3b^2 = cd$.

Let $a,b,c,d$ be positive integers, with $\gcd(c,d)=1$, such that $$a^2+3b^2=cd.$$ By well-known classical results, we have that $c$ and $d$ are both of the form $u^2+3v^2$. QUESTION: Is it valid to ...
0
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1answer
26 views

Lower boundary of quadratic form

I have a quadratic form $x^TAx$ where $x$ is an $n \times 1$ vector and $A$ is a positive definite matrix in the sense that it has only positive eigenvalues. Am I right to say that $||x^TAx|| \ge ...
0
votes
1answer
55 views

Positiveness of a specified quadratic form

The condition of the positiveness of a ordinary quadratic form can be derived by getting the condition of positiveness of a square matrix, like ${v}^{T}{A}{v} \geq 0$ is equal to matrix $A \geq 0$ ...
3
votes
1answer
73 views

How to solve similar with Transformation of a quadratic form into diagonal form?

Define $\color{red}{f=f(x),f'=f'(x)}$,where the derivative with respect to $x$ of a function $f(x)$ is denoted $ f'(x)$. Now give six postive numbers $k_{1},k_{2},k_{3},k_{4},k_{5},k_{6}$, and a ...
2
votes
1answer
31 views

Finding the spinor norm of an element using a proposition in 'The Maximal Subgroups of the Low-Dimensional Finite Classical Groups'.

Let $F=\mathbb{F}_{q}$, where $q$ is an odd prime power. Let $e,f,d$ be a standard basis for the $3$-dimensional orthogonal space $V$, i.e. $(e,e)=(f,f)=(e,d)=(f,d)$ and $(e,f)=(d,d)=1$. I have an ...
3
votes
0answers
34 views

Statement about composition of binary quadratic forms in “A Course in Computational Algebraic Number Theory”

On p.239 A Course in Computational Number Theory, Cohen writes "Although the group structure on ideal classes carries over only to classes of quadratic forms via the maps $\phi_{FI}$ and $\phi_{IF}$ ...
0
votes
1answer
32 views

Representations of some primes as $x^2-2y^2$?

I am looking for (elementary) proofs (idea of the proofs is also OK) or references to proofs of the followings: $$ p\equiv\pm1(\mod8)\longrightarrow p=x^2-2y^2 $$ Any help appreciated.
0
votes
1answer
18 views

Hurwitz's matrix equations

I have a question about the proof of Hurwitz's 1-2-4-8 theorem about the sum of squares. I have consulted Chapter 1 of Rajwade's "Squares" book, notes by Keith Conrad, and notes by Daniel Shapiro. ...