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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0answers
29 views

Find solution set to general bivariate quadratic curve given three points

I know a the function $f(x) = ax^2 + bx + c$ where $ a,b,c \in \mathbb{R} $ can be uniquely defined given three points say $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ via Gaussian elimination. However in the ...
0
votes
6answers
154 views

Describe the rational points on $3x^2 + y^2 = 4$

Apart from $(x, y) = (0, 2)$ and $(1, 1)$, are there any nonzero rational points on the curve $3x^2 + y^2 = 4$ ?
7
votes
1answer
48 views

General definition of angle/ rotation

It is well known that in the Euclidean plane a rotation about the origin can be computed with the formula $$R_{\theta}(x,y) = \big(\cos(\theta)x-\sin(\theta)y, \sin(\theta)x+\cos(\theta)y\big)$$ It ...
11
votes
3answers
541 views

What's the use of quadratic forms?

Starting with the abstract concept of a vector space, I can see why we'd want to add some structure to be able to perform useful operations. For instance if we add a metric/ norm to a vector space we ...
4
votes
4answers
93 views

What are all the concordant forms $n$ such that $a^2+b^2 = c^2,\,a^2+nb^2=d^2$ for $n<1000$?

Part I. The list of congruent numbers $n<10^4$ such that the system, $$a^2-nb^2 = c^2$$ $$a^2+nb^2 = d^2$$ has a solution in the positive integers is known (A003273) $$n = 5, 6, 7, 13, 14, 15, ...
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2answers
39 views

A quadratic form is positive-definite iff its set of isotropic vectors is trivial

Considering a quadratic form $Q$ in a finite dimensional vector space $V$ can I say that $\mathscr{I}=\big\{ \vec{o} \big\} \iff Q $ is definite positive ? Where $\mathscr{I}$ is the isotropic ...
6
votes
2answers
64 views

Is the quadric $3$-fold $v^2 + w^2 + x^2 + y^2 + z^2 = 0$ isomorphic to $P^3$?

The subset of projective $4$-space given by $5$-tuples $[v:w:x:y:z]$ with $v^2 + w^2 + x^2 + y^2 + z^2 = 0$ is birational to projective $3$-space. I think it has the same cohomology as projective $3$-...
2
votes
2answers
97 views

On $p^2 + nq^2 = z^2,\;p^2 - nq^2 = t^2$ and the “congruent number problem”

(Much revised for brevity.) An integer $n$ is a congruent number if there are rationals $a,b,c$ such that, $$a^2+b^2 = c^2\\ \tfrac{1}{2}ab = n$$ or, alternatively, the elliptic curve, $$x^3-n^2x = ...
2
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1answer
17 views

anisotropic Forms over 2-adic integers

I would like to know, if there is a 4 dimensional anisotropic quadratic form over the 2-adic Integers $\mathbb{Z}_2$, that satisfies the following property: It is in diagonal form and 2 does not ...
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0answers
43 views

Real Quadratic Forms, Complex Quadratic forms, and the Inertia Theorem.

I am very confused about the classification of quadratic forms. Scroll to the last paragraph for my question. Here is what I know: A $\bf \text{real}$ quadratic form is obtained from any bilinear ...
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0answers
42 views

Surjective quadratic mapping

Are there any known values of $n$ for which there exists a surjective quadratic mapping $Q:\mathbb{R}^n \rightarrow \mathbb{R}^n$ with non-trivial zeroes?
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2answers
54 views

Complete the square of three variable quadratic expressions

We know that completing $ax^2+bxy+cz^2$ into forms of $k_{1}(a_{1}x+b_{1}y)^2+k_{2}(a_{2}x+b_{2}y)^2$ is easy and have some fixed routine. But the 3 variable case $$ax^2+by^2+cz^2+dxy+exz+fyx$$does ...
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1answer
28 views

About quadric classification by completing square

I'm doing a seminar of geometry. We're learning how to classify quadrics with Maple, and there's a steps we have to follow in order to find what kind of quadric we have. Initially, they give me this: ...
5
votes
1answer
194 views

On products of ternary quadratic forms $\prod_{i=1}^3 (ax_i^2+by_i^2+cz_i^2) = ax_0^2+by_0^2+cz_0^2$

The equation, $$ (ax_1^2+by_1^2)(ax_2^2+by_2^2) = ax_0^2+by_0^2\tag1$$ has the well-known solution when $a=b=1$, $$ (x_1^2+y_1^2)(x_2^2+y_2^2) = (x_1 y_2 + x_2 y_1)^2 + (x_1 x_2 - y_1 y_2)^2$$ ...
2
votes
1answer
68 views

Indefinite Ternary Forms

Consider the indefinite diagonal ternary form $$q(x,y,z)= 2 x^2 + 5 y^2 - 10 z^2$$ Based on numerical experience, I found that any given number of the form 5t+2 is represented either by $q$ or $-q$. ...
-1
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1answer
25 views

solve the following equations for x exists in c

$a)$ $z^3 = (1-i\sqrt{3})8$ $b)$ $z^2 - (3-2i)z + (1-3i) = 0 $ $c)$ $z^4 + 1 + i\sqrt{3} = 0$ I know for the last two you start by using the quadratic form but I'm not sure what to do for any of ...
1
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4answers
61 views

How do you factor $x^2-x-1$?

I know you can't have all integers, but how do you factor this anyway? Wolfram|Alpha gives me $-\frac{1}{4} (1+\sqrt{5}-2 x) (-1+\sqrt{5}+2 x)$. Cymath gives me $(x-\frac{1+\sqrt{5}}{2})(x-\frac{1-\...
1
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0answers
23 views

Quadratic expression equal to zero

Consider the expression $$ A^T V A=0 $$ where $V$ is a $l\times l$ strictly negative definite matrix and $A$ is a $l\times 1$ vector. Is it correct to say (1) $A^T V A=0$ if and only if $A=0_l$, ...
0
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1answer
43 views

Values of quadratic form on unit circle

We have the quadratic form $q(\begin{pmatrix}x\\y\end{pmatrix})=11x^2-16xy-y^2$. Which values does $q$ take on the unit circle $x^2+y^2=1$? I know that $q(x,y)$ is given by $q(x,y)=(x,y)\begin{...
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0answers
25 views

What determines the number of families to $1-4x-4(1-x^2)z = w^2$?

This is related to this post. First, we have, Theorem: "If $w_0, z_0$ is a solution to, $$1-4x-4(1-x^2)z = w^2\tag1$$ then, $$w = w_0+2(x^2-1)n$$ $$z = z_0+w_0\,n+(x^2-1)n^2$$ is also a ...
0
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0answers
20 views

Quadratic Forms over Integers

Given the following quadratic form $a x^2 + b xy + c y^2 + d = 0$ where $a,b,c,d \in \mathbb{Z}$, is there a general method by which I can find $x,y\in \mathbb{Z}$ that satisfy this equation? In ...
0
votes
1answer
46 views

How to solve the quadratic form

I am a physicist and I have a problem solving this \begin{equation} Q(x)=\frac{1}{2}(x,Ax)+(b,x)+c \end{equation} In a book it says that: "The minimum of Q lies at $\bar{x}=-A^{-1}b$ and \begin{...
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2answers
46 views

Fast verification of solution to x'Ax<C

Assume we have some complex vector with N dimensions $\vec x$. We need to verify if this is a valid solution to: $\vec x^HA\vec x<C$ where $A$ is a Hermitian matrix and $C$ is some real constant....
2
votes
0answers
35 views

Can systems of equations of diagonal quadratic forms be solved by Gaussian Elimination

Can the following system of equations be solved using Gaussian Elimination? $$ \begin{bmatrix} s_{00} & s_{01} & s_{02} & s_{03}\\ s_{10} & s_{11} & s_{12}...
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0answers
51 views

Classify the surface $x^2 + y^2 - z^2 + 2xy - 2xz - 2yz - y = 0$

I am working on a problem in which I must classify the surface described by the following equation $$x^2 + y^2 - z^2 + 2xy - 2xz - 2yz - y = 0.$$ I have looked at this Stack Exchange discussion (on ...
4
votes
2answers
105 views

Given $p \equiv q \equiv 1 \pmod 4$, $\left(\frac{p}{q}\right) = 1$, is $N(\eta) = 1$ possible?

Given distinct primes $p$ and $q$, both congruent to $1 \pmod 4$, such that $$\left(\frac{p}{q}\right) = 1$$ and obviously also $$\left(\frac{q}{p}\right) = 1$$ is it possible for the fundamental unit ...
1
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3answers
261 views

Help solving $ax^2+by^2+cz^2+dxy+exz+fzy=0$ where $(x_0,y_0,z_0)$ is a known integral solution

Help solving over the integers: $$ax^2+by^2+cz^2+dxy+exz+fzy=0$$ where $(x_0,y_0,z_0)$ is a known integral solution and $a,b,c,d,e,f$ are integral coefficients. I found in Tito Piezas' identities the ...
3
votes
2answers
72 views

Equivalence of Quadratic Forms that represent the same values

An integer quadratic form is a function $Q(x,y) = ax^2 + bxy + cy^2$ where the numbers $a,b,c \in \mathbb Z$. Call the set of values a quadratic forms takes on $V(Q) = \{ Q(x,y) \in \mathbb Z | x,y \...
2
votes
1answer
32 views

Non-trivial kernel if quadratic form is indefinite

I am wondering the following: if $f:\;V\to \mathbb{R}$ is a quadratic form that is neither positive or negative definite, must its kernel be non-trivial? Here quadratic form means $f(v)=g(v,v)$ where ...
1
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0answers
35 views

Defining a Euclidean Structure on a real vector space.

This comes from a homework question: For $\bf x, \bf y$ $\in \mathbb{R}^n$, put $\langle {\bf x}, {\bf x} \rangle$ = $\sum_{i=1}^n 2x_i^2 - 2\sum_{i=1}^{n-1}x_ix_{i+1}$. Show that the corresponding ...
1
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1answer
38 views

Question about the proof of simultaneous diagonalization of quadratic forms

I'm trying to understand a proof about this theorem and I find myself stuck in a step. Let's consider two quadratic forms $\langle q, Aq \rangle$ and $\langle q, Bq \rangle$, where the first one is ...
6
votes
2answers
47 views

Show $p$ prime s.t. $p \not\equiv 1 \mod 3$ is represented by the binary quadratic equation.

I am working on the following question: Let $p>3$ be a prime such that $p \not\equiv 1 \mod 3$. Show that $p$ is not represented by the binary quadratic equation $f(x, y) = x^2 + xy + y^2$. I ...
0
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0answers
11 views

Connection between maximizing a quadratic form and maximal variance

In order to find the "directions of maximum variance" of $X$ one finds the eigen decomposition of the variance covariance matrix $X^tX$. I have seen the eigenvector problem cast as maximizing the ...
0
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0answers
12 views

Is this the correct way to interpret results from the principal minor test?

Am I correct in thinking that, given all the principal minors $A_{i}$ of the matrix associated with a quadratic form $Q(x,y)$, the Principal Minor Test states that $Q(x,y)$ is Positive Definite if ...
2
votes
1answer
49 views

Prove a binary quadratic equation has specific number of solutions

How do I show that the binary quadratic equation $f(x, y) = x^2 + xy + y^2 = 1$ has exactly $6$ solutions? The discriminant is $-3$, so I cannot use Pell's Equation ($x^2 - dy^2 = p$, where $d>0$ ...
1
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1answer
29 views

When is $SO(m,n)$ simple as a Lie group? What are the Zariski and Euclidean components?

Let $SO(m,n)=\operatorname{SO}(m,n)(\mathbb{R})$ denote the real $(m+n) \times (m+n)$ matrices, with determinant $1$, which preserve the quadratic form $x_1 + \cdots + x_m - x_{m+1} \cdots - x_{m+n}$ ...
1
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1answer
33 views

Quadratic form and spectral theorems

Let P be a quadratic form with real coefficients in $\mathbb{R}^{n}$ such that $P^{-1}(1)$ is non-empty and compact (bounded). Show that there is an orthogonal transformation that maps $P^{-1}(1)$ ...
0
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1answer
23 views

Is $x^TAy$ convex or concave, for $x,y$ not identically equivalent?

It is well known if we had something like: $f(x) = x^TQx$ A quadratic form, is positive semidefinite of $Q$ is positive semidefinite How is the structure of $f(x,y) = x^TQy$ analyzed? i.e. what ...
2
votes
2answers
77 views

Diagonalisation of a quadratic form.

Find a coordinate transformation diagonalizing the quadratic form. Interesting in answering number 2. So, here is my approach:- Step 1:- Write the matrix representation of the equation, that is A= ...
0
votes
2answers
51 views

solution of quadratic equation n unknown

My question is as follows: Let the equation $V^{\top}MV=F$. Such as $V^{\top}=(x_1,x_2,...,x_n)$ a line vector of n unknown coefficients, M a known diagonal matrix (of size n) and F a real number ...
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1answer
50 views

Definition of equivalence of quadratic forms.

I am reading A Course in Arithmetic by J-P Serre. Definition $7$ on page $32$ says Two quadratic forms $f$ and $f'$ are called equivalent if the corresponding modules are isomophic. I am not ...
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3answers
121 views

Writing an expression as a sum of squares

I'd like to write $2xy+2xz+2yz$ in the form $a(\cdots)^2+b(\cdots)^2+c(\cdots)^2$ where each blank space is a linear combination of $x,y,z$. The closest I have is: $$(x+y+z)^2-(x-z)^2-y^2=2xy+4xz+2yz$...
0
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1answer
54 views

Which integer from 1 to 20 does the quadratic form <7,11> over Q represent? [closed]

Which integer from 1 to 20 does the quadratic form $<7,11>=7x^2+11y^2$ over $\mathbb{Q}$ represent? This is an exercise from chapter 1 of Lam's book, Introduction to Quadratic forms over fields. ...
0
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0answers
16 views

Find curves that generate a dual basis to a space of one-forms

Let $V$ be the vector space of one-forms on the plane that have quadratic functions as coefficients of $dx$ and $dy$, with basis $\{x^2dx,xy\;dx,y^2dx,x^2dy,xy\;dy,y^2dy\}$. For any curve $\Gamma$ in ...
1
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1answer
37 views

Quadratic Form - find a minimal scalar $m \in \Bbb R$ such that $q(x,y,z) \le m(x^2+y^2+z^2)$

Let $q (x,y,z)$ be a quadratic form, $$q(x,y,z)=2zx+4yz-2xy $$ $$V=\Bbb R^3$$ Find a minimal scalar $m \in \Bbb R$ such that $$q(x,y,z) \le m(x^2+y^2+z^2)$$ for all $x,y,z \in \Bbb R$. ...
1
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0answers
24 views

Quadratic form of Kroenecker products of skew-symmetric matrices

I am trying to understand under which conditions on $P=P^\top>0$ , $C=C^\top$, the following quadratic form is zero: $$ x^\top \left( D U^\top \frac{L-L^\top}{2} U \otimes PC \right)x = 0 $$ ...
3
votes
2answers
76 views

Bilinear maps and functions of the form $(x,y) \mapsto ux^2 + 2vxy + wy^2$

I was recently reading from the book "Vectors, Pure and Applied", $\S 16.1$ on bilinear forms. It begins In section 8.3 we discussed functions of the form $$(x,y) \mapsto ux^2 + 2vxy + wy^2$$ ...
0
votes
1answer
71 views

Homework question on quadratic forms and change of coordinates

I have been given the following question (on a homework): i) Write down the symmetric matrix A corresponding to the quadratic form $q(v)= wz-xy$ in the 4 variables $w,x,y,z$. I have the matrix A = $...
1
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3answers
65 views

Express a quadratic form in three variable in the format $x^tAx$ using a substitution $x=Py$

I was asked to determine if a quadratic form is positive definite. To do so I must convert in the format $x^tAx$ using a substitution x=Py. So that "it can be written in diagonal form". $$Q(x,y,z) = ...