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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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
20 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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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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0answers
33 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
129 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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18 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
32 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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16 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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0answers
22 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
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2answers
198 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
28 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 ...
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2answers
115 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 ...
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0answers
19 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 ...
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1answer
98 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 individ,...
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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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1answer
25 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, ...
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0answers
59 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 ...
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39 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, ...
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1answer
64 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
106 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 $3x+1$ ...
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1answer
60 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 ...
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1answer
32 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 \...
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1answer
56 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 $A \geq 0$ ...
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0answers
36 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 $\phi_{IF}$ ...
2
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1answer
33 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 ...
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3answers
70 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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2answers
45 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.
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1answer
20 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. ...
3
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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 ...
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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.
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0answers
11 views

Lagrangian of quadratic form with linear constraints

I am working through this paper, and I am confused as to how the author obtains the Lagrangian in equation (1.4) More specifically, in this paper, the authors present a problem: Minimize $x^T A x$ ...
2
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1answer
57 views

Quadratic form.

Consider the quadratic form $Q(v)=v^{t}Av,v=(x,y,z,w)$ where matrix $A$ is given by \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & \\ 0 & 0 & 0 &...
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1answer
120 views

On a remarkable system of fourth powers using $x^4+y^4+(x+y)^4=2z^4$

The problem is to find four integers $a,b,c,d$ such that, $$a^4+b^4+(a+b)^4=2{x_1}^4\\a^4+c^4+(a+c)^4=2{x_2}^4\\a^4+d^4+(a+d)^4=2{\color{blue}{x_3}}^4\\b^4+c^4+(b+c)^4=2{x_4}^4\\b^4+d^4+(b+d)^4=2{x_5}...
2
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1answer
30 views

Relation between Pfister forms in W(Q)

For $n$ a nonzero integer, is it possible to have a relation $$ n\langle\langle-1,-1\rangle\rangle=\sum_i n_i\langle\langle a_i,b_i\rangle\rangle$$ in $I^2(\mathbb Q)\subset W(\mathbb Q)$ (or maybe $...
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1answer
69 views

How canonical is Gauss's law of composition of forms

Gauss defined the composition of binary quadratic forms $f$ and $g$ to be another binary quadratic form $F$ such that there exist integral quadratic forms $$ \begin{align} r(x_0,x_1,y_0,y_1) &= ...
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1answer
48 views

How to write a given element of the orthogonal group as a product of reflections

Let $V$ be a 3-dimensional vector space over a finite field $F$ of $q$ elements, where $q$ is an odd prime power. We know that the orthogonal group $O(V)$ is generated by reflections. How can a given ...
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Maximizing $y^H ( I - X pinv(X) ) y $ with respect to matrix $X$. How hard can it get?

Assume $X$ to be a tall block-diagonal matrix where each block is a collumn vector. Assuming $X^+ = (X^H X)^{-1}X^H $ to be the pseudoinverse of the matrix $X$, find $X$ which maximizes $$y^H ( X X^+ ...
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36 views

Clifford Algebra Isomorphic to Exterior Algebra

Let $E$ be a vector space over a field $k$ and $Q$ be a quadratic form, that is, $$Q:E\to k$$ such that $$Q(\lambda e)=\lambda^2Q(e)\forall\lambda\in k\,e\in E$$ and such that $P_Q:E^2\to k$ is ...
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0answers
43 views

Integral of a multivariate Gaussian distribution over quadratically separated partions

Imagine in the space of $\Re^n$, the quadratic curve $c: f(\mathbf{x}) = \mathbf{x}^TW\mathbf{x} + \mathbf{w}^T\mathbf{x} + w_0$ (with $W$ being a symmetric positive definite matrix, $\mathbf{w}$ a ...
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0answers
13 views

Expectation of quadratic form of correlated variables

Suppose $U$ and $V$ are two $n\times 1$ dependent vectors, in the sense that $E\left( UV^{\prime}\right) \neq \mathbf{0}.$ For a given constant $n\times n$ matrix $A,$ is there is any simple way to ...
4
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1answer
100 views

What are numbers $n$ such that $a^2+nb^2 = c^2$ and $na^2+b^2 = d^2$?

Let $n$ and $a,b,c,d,$ be in the positive integers. I. For the system, $$a^2-nb^2 = c^2\\a^2+nb^2=d^2$$ then $n$ is a congruent number. The sequence starts as $n=5,6,7,13,14,15,20,21,$ and so ...
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53 views

Limit of ratio of quadratic forms

Let $Q_A,Q_B : \mathbb R^n \rightarrow \mathbb R$ be quadratic forms. Find a necessary and sufficient condition for $\lim_{\vec x \rightarrow \vec 0} \frac{Q_A(\vec x)}{Q_B(\vec x)}$ to exist in the ...
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1answer
22 views

Quadratic form inequality implies matrix inequality?

Suppose we have the following quadratic form: $$ x^T(t)(A^TP+PA)x(t)\le-x^T(t)Qx(t)\quad\forall t $$ where $P$ and $Q$ are symmetric positive definite matrices and $\dot x(t)=Ax(t)$. Why does the ...
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1answer
13 views

“Same range of values”, quadratic form transformation

A quadratic form in the variables $u_i$ is expressed as $u'Du$. Matrix $T$ consists of the n characteristic vectors (of matrix $D$): $T = [v_1\quad v_2\quad ...\quad v_n]$. The following ...
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2answers
30 views

matrix trace bilinear form

I'm wondering about this problem on bilinear forms : We have $\phi : \mathbb{M_{n}(R)}*\mathbb{M_{n}(R)} \rightarrow \mathbb{R}$ $$(A,B) \rightarrow trace(AB)$$ I've proved $\phi$ is a bilinear form ...
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0answers
19 views

Reduction of Two Independent Random Variables in Quadratic Form

Consider the $n \times 1$ random vector $\mathbf{x}$ and the $p \times 1$ random vector $\mathbf{y}$. The vectors are independent of each other, and $\mathbf{y}$ has an expected value of zero. I want ...
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0answers
17 views

Stabiliser of a non-isotropic 1-space in $\Omega(n,q)$

Let $n,q$ be odd, $V$ be the $n$-dimensional vector space over $\mathbb{F}_{q}$, and consider the subgroup $$G=\Omega(n,q)=\{r_{v_{1}}r_{v_{2}}\dots r_{v_{k}} : k \textrm{ even }, \prod_{i=1}^{k}{(v_{...
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1answer
45 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 ...