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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2
votes
1answer
92 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 ...
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 ...
3
votes
2answers
168 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 $f: \mathbb{A} \to \mathbb{R} / f \in \mathbb{C}^3 \wedge (P \in \mathbb{A} : \nabla f(P)=0)$. ...
1
vote
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
votes
1answer
58 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 $a,b,c,m_1$ ...
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
4answers
132 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
votes
2answers
113 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
votes
2answers
86 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 ...
0
votes
2answers
142 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 ...
1
vote
1answer
215 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 ...
0
votes
2answers
331 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 ...
3
votes
1answer
122 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 ...
0
votes
1answer
246 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 ...
1
vote
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 ...
2
votes
2answers
180 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 ...
0
votes
1answer
119 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 ...
3
votes
1answer
644 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
votes
2answers
60 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 ...
1
vote
1answer
63 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 ...
0
votes
1answer
177 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
0
votes
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 ...
0
votes
0answers
184 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 ...
0
votes
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 ...
1
vote
0answers
99 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 ...
4
votes
5answers
200 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 ...
0
votes
1answer
54 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 ...
6
votes
0answers
143 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
votes
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 ...
1
vote
2answers
78 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 ...
3
votes
2answers
222 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 ...
5
votes
0answers
98 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
votes
1answer
116 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 ...
0
votes
2answers
37 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 ...
0
votes
1answer
127 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 ...
0
votes
2answers
90 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 $(x-13)(x+1)$ ?
1
vote
0answers
65 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: ...
0
votes
3answers
95 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 ...
7
votes
2answers
291 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 ...
0
votes
1answer
156 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 $D$ ...
0
votes
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 ...
1
vote
0answers
21 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 ...
0
votes
1answer
53 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!
0
votes
1answer
118 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 ...
3
votes
0answers
37 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 ...
1
vote
2answers
34 views

problem with quadratic equation two variable

I have following equation $a^2+4.8ab-b^2=0$ and I have problem with solving it, I don't know why $a=-5 $ or $ a=0.2 $
1
vote
3answers
321 views

How to put $2x^2 + 4xy + 6y^2 + 6x + 2y = 6$ in canonical form

We are given the equation $2x^2 + 4xy + 6y^2 + 6x + 2y = 6$ We did an example of this in class but the equation had less terms. I took a note in class that says : if there are linear terms, I have ...
0
votes
2answers
84 views

Binary quadratic forms whose discriminant is that of a quadratic number field

Let $K$ be a quadratic number field, $D$ its discriminant. Let $ax^2 + bxy + cy^2$ be an integral binary quadratic form such that $D = b^2 - 4ac$. It seems that gcd$(a, b, c) = 1$(see this question). ...
0
votes
1answer
607 views

Quadratic form in canonical form

Reduce the quadratic form $q(x,y) = 6xy$ using the orthogonal reduction (i.e, find a orthogonal basis such that the matrix of the bilinear form is diagonal and $a_{ii} = 0$ or $a_{ii} = ^+_-1$) What ...
0
votes
1answer
153 views

Hyperbolic lattice and its cone

By lattice we mean a finitely generated free abelian group $L$ equipped with an integral non-degenerate symmetric bilinear form $L\times L\rightarrow\Bbb{Z}, \ (x,y)\mapsto x\cdot y$. We call $L$ ...