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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Under What Intervals Is A Matrix Positive Definite, Positive Semi-Definite, Indefinite, Negative Definite and Negative Semi-Definite? [on hold]

Suppose we have a matrix which represents a quadratic form. $$ \begin{matrix} a & -a & -3a \\ -a & 2a & 2a \\ -3a & 2a & (9a+2) \\ ...
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13 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 ...
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0answers
8 views

Small roots of multivariate polynomials

I'm looking for a program/algorithm (or even "theory"?) which checks if a given multivariate polynomial, say $P(x_1, \dots, x_n)$ has a \textit{real} root in some given region, say a closed ...
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1answer
17 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
12 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
16 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
13 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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13 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 }, ...
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1answer
34 views

Degenerate quadratic form

I'm beggining with quadratics forms and I ma wondering : Let $a$ be a real number and $q:\mathbb{R^4} \rightarrow \mathbb{R}$ $(x,y,z,t) \rightarrow ax^2+2axy+y^2+4zt-at^2$ I would like to know for ...
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2answers
30 views

Maximums on Quadratic Functions [closed]

How do you find the maximum of a quadratic function? Specifically, $R(x) = -4x^2 + 4000x$
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1answer
27 views

A two-variable quadratic form over a field of characteristic 2 with no nontrivial roots

I'm looking for a quadratic form of the form $q(x,y)=ax^2 + bxy + cy^2 \in F[x,y]$, where $F$ has characteristic 2, and $q(x,y)$ has no roots besides the obvious one, $x=y=0$. I've proved the case of ...
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3answers
28 views

A nontrivial solution to the quadratic form $x^2 - xy + y^2$ over the finite field $𝔽_p$ with $p ≡ 1 \pmod3$ a prime

I'm trying to prove that when $p ≡ 1 \pmod3$ is a prime, $p$ is reducible over the Eisenstein integers, and I've gotten to the point where, provided $p\,|\,u^2 - u + 1$ for some integer $u$, then $p$ ...
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1answer
36 views

Is the squared euclidean norm a measure for the distance of two points?

I like to prove that a measure for the distance $d$ of two points $\vec a$ and $\vec b$ in $R^N$ is given by the squared euclidean norm $$d^2= \sum^N_j (a_j - b_j)^2 $$ So far I was able to show ...
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1answer
27 views

Vertex Form of Parabola - Why does it work?

Recently, I have been trying to plot parabolas of quadratic equations. First, I have to convert them to vertex form and then we can easily plot them. This makes me wonder why the vertex form of a ...
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0answers
49 views

Subtle error with a module endormorphism on $\mathbb{Z}_8 \times \mathbb{Z}_8$

Let $a,b,c$ be arbitrary integers such that $a$ is odd and $(a,b,c)=1$. Let $R = \mathbb{Z}_8$, the set of all integer residues modulo $8$. Define an $R$-module endomorphism $\phi \colon R \times R$ ...
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0answers
42 views

Expectation of a special form of quadratic form

Let $\mathbf x$ be a $n\times1$ random variable, $\mathbf s$ be a vector of size $3\times 1$ and $A$, $M_1$, $M_2$ and $M_3$ be $n\times n$ matrices. What is the following expectation with respect to ...
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0answers
19 views

Reduction of a quadratic form to a canonical form

I'm supposed to reduce following polynomial to its canonical form. But my result differs from the one given in my book, so I'm not sure if it's correct too. $$ q = u_{xx} - u_{xy} - 2 u_{yy} + u_x + ...
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0answers
25 views

$\det$ is the only multiplicative nonzero quadratic form on $\mathcal M_2(\Bbb R)$

Let $q$ a nonzero quadratic form on $\mathcal M_2(\Bbb R)$ verifying the relation $$\forall A,B\in\mathcal M_2(\Bbb R),\; q(AB)=q(A)q(B)$$ The question is to prove that $q=\det$. What I have tried ...
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1answer
10 views

Hint on proving that the component in a radical splitting of a quadratic space is regular

I'm stuck on the following exercise from Basic Quadratic Forms by Larry Gerstein. In a radical splitting $V = \mbox{rad} V \perp V_1$, show that $V_1$ is regular. I want to let $v \in \mbox{rad} ...
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2answers
53 views

Equation of a cone

Find the equation of the cone whose vertex is at the origin and whose directing curve is given by the equations: $$\begin{cases} x^2-2z+1=0 \\ y-z+1=0\end{cases} $$ We know that an eliptic cone is ...
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0answers
8 views

Decompose matrix into directions with uniform variance

Singular Values Decomposition (SVD) can be viewed as decomposing a matrix $M\in\mathbb{R}^{N\times M}$ into directions such that for any $k=1..N$ the k first directions capture the largest amount of ...
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0answers
15 views

Distribution of the norm of uniform random unit vector after linear transformation

Suppose that $\mathbf{u}$ is a uniform unit vector. It is obtained as $\mathbf{u}=\frac{\mathbf{n}}{||\mathbf{n}||}$ where $\mathbf{n}$ is a white Gaussian vector. Clearly we have ...
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1answer
52 views

Minimum value of $\frac{(1 + x + x^2)(1 + y + y^2)}{xy}$

What is the minimum value of $$\frac{(1 + x + x^2)(1 + y + y^2)}{xy},~~(x \neq 0)$$ Should we find the minimum value of each quadratic?
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0answers
37 views

A function of two quadratic forms

Given two functions $f(\mathbf{x^TAx})$ and $g(\mathbf{x^TBx})$, consider the new function $h(\mathbf{x})=f(\mathbf{x^TAx})g(\mathbf{x^TBx})$. Can $h$ be a function with a quadratic argument of the ...
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1answer
12 views

How to decide range of a quadratic form under constraint condition?

Is there any easy way to decide range of $x_1^2 - 2x_2^2 + x_3^2 + 2{x_1}{x_2} - 4{x_1}{x_3} + 2{x_2}{x_3}$ under $x_1^2 + x_2^2 + x_3^2 = 1$? I tried with calculus and found it a bit difficult. Can ...
2
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1answer
40 views

expressing a quadratic map as a complex map

Are there any known criterion when a real quadratic mapping $ Q:\mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n} $ can be expressed as a complex quadratic map $ Q:\mathbb{C}^n \rightarrow \mathbb{C}^n$? ...
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0answers
37 views

Looking for a proof of a known theorem involving integral quadratic forms

Let $n$ be a positive integer and let $Q$ be an integral quadratic form in $n$ variables. Let $M$ be the symmetric "two's in" matrix associated with $Q$ so that $Q$ can be expressed as the $1 \times ...
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2answers
30 views

Using Lagrange's diagonalization on degenerate linear forms

Let $A=\begin{pmatrix}1 & 2 & 3\\ 2 & 3 & 4\\ 3 & 4 & 5 \end{pmatrix}$ be a real matrix. Find an invertible matrix $P\in M_{3}(\mathbb{R})$ such that $P^TAP$ is diagonal ...
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1answer
25 views

Positive definite binary quadratic Forms

Please help me to solve this question or introduce references that help me: Let $f(x,y) = ax^2 + bxy + cy^2$ be a reduced positive definite form. Suppose that $g.c.d(x, y) = 1$ and that $f(x, y) ≤ ...
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44 views

The group defined by Gauss's definition of composition of forms

In article 242 of Disquisitiones, Gauss investigates the properties of the direct composition of two forms of the same discriminant. In this case, he gives a "natural" choice for such a composition. ...
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1answer
24 views

Primitive Integral Quadratic forms of fixed discriminant

Assume all the quadratic forms below are integral. Use your favorite definition of discriminant of a quadratic form(a rational multiple of a matrix associated to the coefficients of the quadratic ...
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0answers
22 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 ...
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6answers
149 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 $3x^2 + y^2 = 4$ ?
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1answer
42 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 ...
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3answers
504 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
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2answers
52 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
32 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 ...
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2answers
59 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 ...
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2answers
88 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
15 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
31 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
38 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
38 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
26 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: ...
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1answer
142 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
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1answer
63 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 $-q$. ...
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1answer
22 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 ...
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4answers
58 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 ...
1
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0answers
20 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 $A=0_l$, ...