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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102 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 ...
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1answer
29 views

for each of the following solve for x and y

Question 1- For each of the following equations 1.1 Solve for x $$x^2-2xy+y^2=0$$ $$5x^2-3xy-8y^2$$ $$8x^2-5xy-13xy^2=0$$
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0answers
66 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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2answers
86 views

Linear Algebra Quadratic True False

"Every quadratic form $x^TAx$ with $A$ an invertible matrix is either positive definite, negative definite, or indefinite." Is this true or false? I am just wondering does it have to be positive, ...
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1answer
106 views

Signature of quadratic form and eigenvalues

I'm asking about the signature of the quadratic form - the triple (n0, n+, n−). Is it true that n+ is the number of positive eigenvalues, and n- is the number of negative of eigenvalues of the matrix ...
2
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1answer
88 views

Quadratic form and symmetric bi linear form formula, basic point unclear to me

Something really basic but I have to ask it: I was taught that the formula of symmetric bi linear form of the quadratic form if: $f(v,w) = 1/2(q(v+w)-q(v)-q(w))$ but $q$ is linear so what did we get ...
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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 ...
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2answers
162 views

On the hessian matrix and relative minima

I'm asked to prove the following statement: Let $\mathbb{A} \subseteq \mathbb{R}^n$ an open subset and $f: \mathbb{A} \to \mathbb{R} / f \in \mathbb{C}^3 \wedge (P \in \mathbb{A} : \nabla f(P)=0)$. ...
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0answers
80 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 ...
3
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1answer
56 views

About the roots of a quadratic equation

Let $m_1$ and $m_2$ the real and diferent roots of the quadratic equation $ax^2+bx+c=0$. Do you know some way to write $m_1^k + m_2^k$ in a simplest form (linear, for example) using just $a,b,c,m_1$ ...
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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 ...
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4answers
125 views

Find $\frac{a^3}{a^6 + 1}$ given a is a root of a quadratic equation

My question is: If a is a root of the equation $x^2 - 3x + 1 = 0$, then find the value of $\frac{a^3}{a^6 + 1}$. So, I figured we can use the Sridharacharya ...
4
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2answers
112 views

If $3\mid a,b,c$ and $n=a^2+b^2+c^2$, prove that there exist $x,y,z$ such that $n=x^2+y^2+z^2$, where $3\nmid x,y,z$.

If $3\mid a,b,c$ and $n=a^2+b^2+c^2$, prove that there exist $x,y,z$ such that $n=x^2+y^2+z^2$, where $3\nmid x,y,z$. Here $n\in\mathbb N$, $a,b,c,x,y,z\in\mathbb Z$. This problem is originally ...
2
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2answers
81 views

Is this expression a quadratic form

I have an matrix expression that basically is of the form: \begin{equation} tr(B X BX ) \end{equation} Where $B$ and $X$ and nonsquare matrices. $B$ is $p \times n$, $X$ is $n \times p$. It ...
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2answers
128 views

Congruent diagonal matrix

For two days I reflect on this question without an answer: If $A=(i+j-1)_{1\le i,j\le n}$ is matrix in $\mathcal M_n(\mathbb R)$, the question is to find basis in which $A$ is congruent to diagonal ...
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1answer
203 views

Let $W \subset V$ vector spaces, let $W'$ be the orthogonal complement of $W$. Show $\dim(W)+\dim(W')=\dim V$

As title says: Let $W \subset V$ vector spaces, let $W'$ be the orthogonal complement of $W$. Show $\dim(W)+\dim(W')=\dim V$. We are given that $W \subset V$ finite vector spaces, symmetric bilinear ...
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2answers
294 views

Find the symmetric matrix that represents the quadratic form $Q(X)=trace(X^2)$, $X\in mat_n\mathbb (R)$

as the title says, find the symmetric matrix (or signature) of $Q(X)=trace(X^2)$ where $X$ is an $n$ by $n$ matrix with real entries. the diagonal of $X^2$ is $$\sum_{k=1}^n x_{ik}x_{ki}$$ So ...
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1answer
120 views

Signature of quadratic form $Q(p)=p(1)p(2)+p(3)p(4)$

I was asked to find the signature of the quadtratic form $Q(p)=p(1)p(2)+p(3)p(4)$ where $p$ is a polynomial in $\mathbb R_n[x]$ I tried doing it via finding the symmetric matrix that $Q$ corresponds ...
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1answer
188 views

Find the signature of the quadratic form

Very simple question but something doesn't make sense to me. We are given a quadratic form (bilinear map but on the same vector twice): $Q(v) = v^t *\begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 ...
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1answer
49 views

Showing that the quadratic form $Q(x,y,z,t)=x^2+y^2+z^2-7\cdot t^2$ is anisotropic on $\Bbb Q^4$

I'm looking for help in order to find a prove that the quadratic form $Q(x,y,z,t)=x^2+y^2+z^2-7\cdot t^2$ on $\mathbb Q^4$ can or cannot take the value $0$ on a nonzero element of $\Bbb Q^4$. I was ...
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2answers
154 views

Is it true that the whole space is the direct sum of a subspace and its orthogonal space?

Problem The ground field is $K$, $\operatorname{char}K\neq2$. Suppose $W$ is a (maybe infinite dimensional) subspace of a vector space $V$ with a symmetric/symplectic form ...
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1answer
101 views

Finding diagonal transformation matrix of a bilinear form

Let $f:\mathbb R^3 \times \mathbb R^3 \rightarrow \mathbb R$ be a symmetric bilinear form, and let $q$ be its quadric form, so that $q(x, y, z)= xy+yz$. Find the transformation matrix $A$ of $f$ by ...
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0answers
45 views

Why is the sign of the determinant of a quadric in $\mathbb R^3$ invariant under affine transformations?

According to my reference, the answer has to do with the fact that the projective closure of a quadric in $\mathbb R^3$ is given by a matrix $\bar A$ of order even (in fact, order 4). But isn't it ...
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1answer
511 views

Polar form of quadratic equations

I'm trying to derive a polar, general and graphing, form of a quadratic equation. Here Is what I've done so far. $$ f(x)=ax^2+bx+c $$ And $$ f(x)=a(x-h)^2+k $$ Then I substituted $$ x=r\cos(\theta) ...
0
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2answers
57 views

Identify a quadric

Could you tell me how to identify a given quadric? Given a conic section, I should find an orthonormal affine frame in $\mathbb{R}^2$ (with standard dot product) in which the equation has a canonical ...
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1answer
62 views

Associated Bilinear Form to Q (Quadratic Form)

I need to diagonalize the quadratic form $Q(x) = {x_{1}}^{2} + 2x_{1}x_{2} + 2{x_{2}}^{2} + 2x_{2}x_{3} + {x_{3}}^{2}$ so I know I need to find the associated Bilinear form with $B(x,x) = Q(x)$ - the ...
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1answer
141 views

Quadratic formula / stationary points

For the question find and classify the stationary points of f(x) Given the function f(x) = ln(x^2 - 2x + 2) Are my calculations right in thinking x = 3.75, -0.75 ? Cheers
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1answer
67 views

Elliptical polarisation

In physic context one find the curve with parametrisation in t, $x=x_0\cos(t)$ and $y=y_0\cos(t+\varphi)$ with is an ellipse with equation ...
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0answers
171 views

selecting a good upper bound on quadratic form in presence of unknown PD matrix

I have a cost function that is \begin{equation} J=\text{Trace} [\ (\ I-LC)\ KQK^T(\ I-LC)\ ^T ]\ \end{equation} where $Q$ is a unknown positive definite matrix, $K$ and $C$ are full rank $n\times ...
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2answers
51 views

How I can find the matrix $A$ of this quadratic form?

Let $(e_1,\ldots,e_n)$ the standard basis of $\mathbb R^n$ and we consider the quadratic form $$\Phi(x)=\sum_{1\le i<j\le n}(x_i-x_j)^2$$ How I can find the matrix $A$ of this quadratic form? My ...
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0answers
95 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 ...
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5answers
193 views

Looking for proof of no solution to 4-variable quadratic diophantine equation

Show there are no integers $a,b,c,d$ such that $$\begin{cases}1=9ac+3ad+3bc-16bd \\ 1=3ad+3bc+2bd \end{cases}$$ Motivation: The ideal $I=(3,1+\sqrt{-17})$ in $R=\mathbb{Z}[\sqrt{-17}]$ has the ...
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1answer
51 views

Finding affine transformation

Find affine transformation which takes the ellipse $x^2+4y^2+2x-8y+3=0$ to the form of the ellipse ${x^2 \over 9}+{y^2 \over 16}=1$. So I took the quadric and reached to a standard form: ${(x+1)^2 ...
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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 ...
5
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1answer
110 views

Why the Little Methuselah form is the “Little Methuselah”s form?

This is my first question on MathStackexchange. Let me know if I am violating rules, or my question is somewhat ugly. I am reading Conway's book "Sensual (Quadratic) Form". He introduces a tenary ...
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2answers
69 views

Convex set, quadratic form

I'm trying to answer a question concerning convex sets "Does the following constraint system define a convex set? $x^T Qx ≤ 1$ $a^T x = 0$ Here, Q is a symmetric and positive definite matrix and a ...
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2answers
206 views

Should diffeomorphisms preserving arc length be affine?

Problem Suppose $\varphi\colon V=\mathbb R^n\to V$ be a differmorphism and $d\varphi$ is its tangent mapping. $\langle\circ,\circ\rangle$ is a nondegenerate (symmetric or symplectic) bilinear form on ...
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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: $\mathbf{x}_{0}$ = $\mathbf{x}_{1}$ + ...
3
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1answer
113 views

Can the equation $ax^2+by^2=cz^2$ be solved in integers (excluding trivial solutions)?

Suppose $a,b,c\in\mathbb{N}$ and are each squarefree. Is there a general solution for this equation? I found that for this equation to be soluble in integers there are three necessary and sufficient ...
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2answers
36 views

Definite Quadratic Form

I got the following problem: Let $V$ be a real vector space and let $q: V \to \mathbb R$ be a real quadratic form, Prove that if the set $L = \{v \in V | q(v) \ge 0\}$ forms a subspace of $V$ then q ...
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1answer
117 views

Quadratic Forms and their Matrices.

1) How do you manage to transform a matrix from quadratic to canonical form? For instance, assume a linear transformation such that: $$Q(x,y,z)=x^2+2xz+z^2;$$ As far as I can see, in the ...
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2answers
83 views

Quadratic Equation to its binomial form

How do I convert a quadratic equation to its binomial form For example how does $x^2 - 12x - 13$ become $(x-13)(x+1)$ ?
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0answers
61 views

Integrate the ratio of quadratic forms

Please, help me to solve the folowing problem. Given two positive-definite $n$-dimensional matrices $A$ and $B$, need to integrate its ratio over unit ball: ...
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3answers
93 views

Additional solutions to quadratic equations which don't match the formula answer.

I'm hoping for link to some resource which can explain why the following is true. $$ x^2 + 104x - 896 = 0 $$ Using the quadratic formula we pull a = 1, b = 104, c = 896. Putting that into the ...
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2answers
288 views

How find this matrix $A=(\sqrt{i^2+j^2})$ eigenvalue

let the matrix $$A=(a_{ij})_{n\times n}$$ where $$a_{ij}=\sqrt{i^2+j^2}$$ Question: Find the difference $sign{(A)}$ can see this define:http://en.wikipedia.org/wiki/Sylvester's_law_of_inertia My ...
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1answer
135 views

Solution to a quadratic form

I'm trying to find a closed form solution of the following quadratic form for $x$. $x^{T}Dx = c$ where $c$ is just a constant placeholder for some terms on the other side. I know that, because $D$ ...
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0answers
96 views

How checking $(-1)^n$ for $H=H_{n+m}$ is equivalent to checking $(-1)^{m+1}$ for $H_{2m+1}$?

This is from "Mathematics from Economists" by Simon and Blume: To determine the definiteness of a quadratic form of $n$ variables, $Q(\mathbf{x})=\mathbf{x}^TA\mathbf{x},$ when restricted to a ...
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0answers
20 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 ...
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1answer
52 views

Where does the “inertia” term come from? [duplicate]

Where does the "inertia" term in regard to quadratic forms (e.g. Sylvester's law of inertia) come from? Thank you!
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1answer
100 views

Is $B$ a positive or negative semidefinite?

Let $A$ be an $n\times n$ symmetric matrix. Then, $A$ is a positive semidefinite iff every principal minor of $A$ is $\geq0$; $A$ is a negative semidefinite iff every principal minor of odd order ...