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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2
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1answer
16 views

Is the restriction of a Minkowski-form in $\Bbb R^n$ on a vector subspace $U$ with $\dim(U) = n - 1$ also a Minkowski-form?

Task: Is the restriction of a Minkowski-form in $\Bbb R^n$ on a vector subspace $U$ with $\dim(U) = n - 1$ also a Minkowski-form? Solution: Since a Minkowski-form has the type $(n - 1, 1)$, ...
1
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2answers
26 views

What are some examples of applications of integral quadratic forms in $n$ variables in algebraic topology?

I'm reading the wiki page of qudratic forms. It simply seems curious to me what are some concrete examples of applications of integral quadratic forms in algebraic topology. I've searched a bit but a ...
0
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1answer
33 views

Confusion of a formula about Lagrangian

Recently, I am reading a paper about eigenvalue problems. Consider the following problem, which occurs at the first page of the paper. \begin{align} \text{minimize}\quad &x^TAx \\ \text{subject ...
3
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2answers
97 views

What do mathematicians mean when they say “form”?

As in differential form, modular form, quadratic form? I'm sorry if this is a really silly question.
0
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1answer
29 views

Reducing a rational ternary quadratic with zeros in its diagonal into a canonical form.

Let the following be a ternary quadratic form: $$A = \begin{pmatrix}0 & a & b \\ a & 0 & c \\ b & c & 0 \end{pmatrix}$$ with $a,b,c\in\mathbb{Q}$. If at least one term in the ...
0
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0answers
30 views

Can the 290 Theorem be refined/sharpened to include special conditions?

The 290 theorem states If a positive-definite quadratic form with integer coefficients represents the twenty-nine integers $1$, $2$, $3$, $5$, $6$, $7$, $10$, $13$, $14$, $15$, $17$, $19$, $21$, ...
3
votes
1answer
64 views

Is this number positive?

Let $(a_{ij})$ be a collection of non-negative numbers indexed by integers $1\le i,j \le N$ where $N$ is some fixed integer. Let $(c_{ij})$ be another collection of real numbers also indexed by ...
0
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3answers
37 views

If $f,g$ are quadratic forms over $\mathbb{R}$ and $f$ is positive definite, can you reduce the both simultaneously to sum of squares?

If $f,g$ are quadratic forms over $\mathbb{R}$ and $f$ is positive definite, can you reduce the both simultaneously to sum of squares? This question appeared from a friend of mine and I did not ...
3
votes
1answer
44 views

How two find the matrix of a quadratic form?

I was wondering. If I have a bilinear symmetric form, it is easy to find its matrix. But, when I have a quadratic form, which is the procedure to do that? I heard that one possibility is: If $q$ is ...
3
votes
1answer
56 views

Irreducibility of a quadric

I am struggling with a problem in Shafarevich's Basic Algebraic Geometry. First, some context: Fix $k$ an algebraically closed field. Lines in $\mathbb{P}^3$ correspond to planes through the origin in ...
0
votes
1answer
32 views

can $4^{2n }$ be written as the sum of “three squares”?

Lagrange theorem says only numbers $n \neq 4^n ( 8k+7)$ can be written as the sum of three squares. what about this one? $$ 4= 2^2 + 0^2+ 0^2 $$ this looks acceptable to me, and yet it is ...
1
vote
1answer
36 views

aggregate two quadratic functions

I have a quadratic function$$W(x_1,x_2,\ldots,x_n)=A\sum_{i} x_i^2+ \sum_{i\neq j} x_ix_j.$$ Denote the input vector as $\textbf{x}$, in quadratic form, $W(\textbf{x})=\textbf{x}^TM\textbf{x}$, where $...
0
votes
0answers
28 views

Linear transformation of positive definite diagonal matrix

Let $\mathbf \Psi$ denote the set of all positive definite, diagonal, nXn dimensional, real-valued matrices . Let $\mathbf \Phi$ denote the set of all positive semi-definite, diagonal, nXn dimensional,...
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0answers
30 views

Is sum (convex combination) of quadratic function/aggregator quadratic?

We know convex combination of concave/convex functions are concave/convex. While convex combination of two quasi-convex/quasi-concave functions necessarily quasi-convex/quasi-concave. Common ...
0
votes
1answer
31 views

Show quadratic form is positive definite

If the quadratic form of $Q(x)=x^T Ax$ and $\langle x,y\rangle=\frac{1}{2}[Q(x+y)-Q(x)-Q(y)]$. Where $x, y$ are vectors in $\def\R{\Bbb R}\R^n$ and $A$ is a $n\times n$ matrix. Show that $\langle\, , \...
1
vote
1answer
64 views

show the quadratic function $W(x_1,x_2,\ldots,x_n)=A\sum_{i} x_i^2+ \sum_{i\neq j} x_ix_j$ is quasi-concave

I have a quadratic function $W(x_1,x_2,\ldots,x_n)=A\sum_{i} x_i^2+ \sum_{i\neq j} x_ix_j$, with $x_i$ nonnegative and $A \in[0,1)$. And w.l.o.g. we can normalize $x_i's$ to between 0 and 1. In ...
1
vote
1answer
29 views

a Matrix that returns the average of a vector?

Let's concentrate in the 2 dimensional case. I'm looking for a $2 \times 2$ Matrix $A$ that for a vector $x=(x_1, x_2)$ will satisfy: $$ xAx^t = \frac{x_1+x_2}{2}$$ Using simple methods of ...
2
votes
1answer
34 views

System of two quadratic equations in two variables with two parameters leads to quintic polynomial

Actually, it's two closely related systems. Let $a,b \in \mathbb{Q}$ be the parameters. The first system has the form: $$(1+a y)x^2-2(a+y)x+(1+a y)=0 \\ (1-b x)y^2-2(b-x)y+(1-b x)=0$$ One of the ...
2
votes
1answer
47 views

Finding the parameters of an ellipsoid given its quadratic form

Suppose we have the quadratic form of an ellipsoid of the form $$ax^2 + by^2+cz^2+dxy+eyz+fxz+gx+hy+iz+j=0$$ I want to find centroid of the arbitrarily oriented ellipsoid, its semi-axes, and the ...
1
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0answers
23 views

Constrained optimization problem using Largange multipliers: ellipsoid collision detection and response

This one is purely for the mathematics so the result is far less important than the method itself. My task is to implement a fast and efficient ellipsoid collision detection and response algorithm. ...
0
votes
1answer
23 views

Partial Derivative of a quadratic form

I want to derive, w.r.t $x$, this: $x'Ax+2y'B'x+y'Cy$ The reference says: "Assuming $A$ positive definite, then the partial derivative is: $2(Ax+By)$." Why the transpose $x'$ it's not in the ...
0
votes
1answer
17 views

condition for a binary quadratic form to be positive at infinity

I have a sort of binary quadratic form $Q(x,y)=\lambda_1x^4 + \lambda_2 x^2y^2 +\lambda_3 y^4$. I can assume $\lambda_1, \lambda_3 > 0$. Consider $$\int dxdy \; e^{-Q(x,y)}$$ What minimal ...
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0answers
26 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: ...
1
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1answer
42 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}) = \...
2
votes
2answers
99 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 ...
0
votes
1answer
44 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 ...
0
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0answers
17 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 ...
1
vote
2answers
43 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 \...
2
votes
3answers
49 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^...
0
votes
1answer
27 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 $...
1
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2answers
47 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
48 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^...
2
votes
4answers
159 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?
3
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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=-\...
0
votes
1answer
31 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, ..., ...
0
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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 ...
3
votes
1answer
71 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 $$x^2-my^2=zw$$ ...
1
vote
1answer
36 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 ($A\vec{v} = \vec{w}$) ...
0
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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 $b$?
1
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1answer
90 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, ...
3
votes
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 ...
0
votes
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: $x_1\cdot x_2 + x_1\cdot x_3 + x_2\cdot x_3$
0
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0answers
33 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 ...
1
vote
2answers
61 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
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 n\...
0
votes
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+...
1
vote
2answers
57 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
votes
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 ...
4
votes
1answer
49 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 ...
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 ...