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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1answer
15 views

Maximization of quadratic form over complex unit cube

I am trying to find the maximum of a hermitian positive definite quadratic form $xQx^H$ (where $Q=Q^H$ and all eigenvalues of $Q$ are non-negative) over the complex unit cube $|x_i|\leq 1$, ...
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0answers
20 views

Differentiating quadratic form containing vector raised to powers elementwise, can I avoid Hadamard notation?

Say $\mathbf{M}$ is a symmetric, p.d. 2x2 matrix, and $\mathbf{x}$ is a 2x1 vector. The familiar quadratic form is of course given by: $A=\mathbf{x'}\mathbf{M}\mathbf{x}$ (where $A$ is a scalar), and ...
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0answers
19 views

existance of a solution to quadratic form equation

Let $\lambda$ is an unknown scalar and; $Q=Q_1 - \lambda*Q_2$ where $Q_1, Q_2$ are $NxN$ positive symetric matrices, $B=B_1 - \lambda*B_2$, where $B_1, B_2$ are $Nx1$ vectors, $m=m_1 - ...
0
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2answers
31 views

Minimum quadratic form value within a line?

If I have $x\in R^n , C\in R^{m\times n}, d\in R^m$, $m<n$, then $Cx=d$ is a linear manifold. And $P\in R^{n\times n}$, $P>0$, the quadratic form is $y=x^TPx$ Is there an analytical expression ...
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0answers
25 views

Semidefinite relaxation of Quadratic equation

I have read in various papers that we can write a Quadratic equation with symmetric matrix as a linear programming problem. For example $$f(x)= x^T*Q*x + c$$ where Q=[2 0;0 3]; Now we can write ...
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0answers
11 views

Reduction of positive definite binary quadratic forms over congruence subgroups

Let $\Gamma_0(N)$ be a congruence subgroup of $\mathrm{SL}_2(\mathbb{Z})$ and $Q(x,y)$ be a positive definite binary quadratic form with leading coefficient $a$ divisible by $N$. Can someone give me a ...
2
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2answers
75 views

Regular Quadratic Space - isotrope vector

I am currently trying to solve the following exercise: Show that every regular quadratic space of finite dimension $E$ that contains at least one isotrope vector, has a basis consisting only of ...
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0answers
8 views

non-uniform arithmetic lattice in a semisimple algebraic group over a local field of positive characteristic

Say I'm considering the group of rational points $G(k)$ where $G$ is the special orthogonal group for the quadratic form $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2}-x_{5}^{2}$ and $k$ is a ...
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0answers
16 views

example of a quadratic form

Would someone be able to tell me an example of a quadratic form defined on a five-dimensional vector space $V$ over a non-archimedean local field $k$ of positive characteristic (not equal to two, say) ...
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0answers
19 views

What is binary norm of quadratic fields of sum of two squares such that one of them is necessarily even like $a^2 +4b^2?$

I am trying to simplify an expression which I have reached, suppose a number can be represented in the form of $D=a^2 + 4b^2$. What is binary norm of $D$, or how else can it be represented?
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0answers
5 views

meaning of index of a quadratic form for a field of positive characteristic

In "Classification of Algebraic Semisimple Groups" in "Algebraic Groups and Discontinous Subgroups: Procedings of Symposia in Pure Mathematics, Volume IX", Jacques Tits speaks of the index of a ...
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0answers
7 views

Perturbed quadratic form definiteness

I have a rectangular matrix $[\mathbf \Pi]\in \mathbb R^{m\times n}$, where $m>n$, and $\mathrm {rank}([\mathbf \Pi])=k<n$. Its left null space basis are the columns of $[\mathbf Z]$. It's ...
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1answer
33 views

Tricky problem about quadratic forms

Let A be symmetric matrix of order 3. Consider set $S=\{x\in\mathbb{R}^{3}:\; x^{T}Ax=a\}$. 1 (true). If S is unbounded for any $a\in\mathbb{R}$, then A is indefinite. 2 (false). If S ...
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0answers
30 views

Hermitian form of a unitary space

If $V$ is a finite $n$-dimensional complex vector space with a hermitian form $h$ then $h$ is given by a hermitian matrix $A$ with the transformation law $P^{t}A\bar{P}$ where $P$ is an invertible ...
0
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1answer
13 views

Diagonalizing quadratic forms

I know how to diagonalize a given quadratic form using the Gaussian method. Though, I once read somewhere that there's another method which uses an augmented matrix, but I didn't go into details. I ...
2
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2answers
30 views

find real numbers so that a specific linear isometry exists

Consider the quadratic form $$f: \mathbb{R}^3 \to \mathbb{R}, (x_1, x_2, x_3) \mapsto 3x_1^2 - 3x_2^2 + x_3^2-2x_1x_3$$ I want to find $\lambda_1, \lambda_2, \lambda_3 \in \mathbb{R}$, so that there ...
8
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1answer
75 views

Why does the plot of the legendre symbol of $x^2 - y^2$ over a finite field look rectangular

The small top-left thing is a plot of the legendre symbol of $x^2 - y^2$ over $\Bbb F_{37}$. The thing in the middle is plot for $\Bbb F_{587}$. The thing on the right is a plot of the legendre ...
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0answers
10 views

Critical points of quadratic form

Consider the problem $$ \text{max } Q(\mathbf{x})=\gamma_1x_1^2 + \gamma_2x_2^2 + \dotsb + \gamma_mx_m^2 \quad \text{ subject to } x_1^2 + x_2^2 + \dotsb + x_m^2 = 1 . $$ The $\gamma$'s are known and ...
0
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1answer
13 views

Show that $|Q(a,b)| \leqslant M \cdot \max(|a|^2,|b|^2).$

Prove if $Q(x,y)=Ax^2+Bxy+Cy^2$ is a quadratic form, then the constant $M = |A|+|B|+|C|$ satisfies the property that $|Q(a,b)| \leqslant M \cdot \max(|a|^2,|b|^2)$ for every $a,b \in \mathbb{R}.$ ...
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0answers
15 views

Quadratic Form for Matrix with Sum

I just had a pretty basic question. Suppose you have the quadratic form: $$(u+x)^t V (u+x)$$ If you expand this out, you get something along the lines of $$u^t V u + x^t V x + u^t V x + x^t V u$$ ...
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0answers
35 views

Is the $\mathbb{Q}_2$- Space$ (\mathbb{Q}_2[\zeta], trace(cxy))$ hyperbolic?

I am working at a Problem for some time and it comes down to the question: Let $K:=\mathbb{Q}_2[\zeta]$ be a cyclotomic extention of the dyadic field $\mathbb{Q}_2$. For any $c \in ...
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0answers
22 views

Cycles of binary forms with coefficients in $F_q^*[T]$

I aim to study the binary forms $ax^2 + bxy + cy^2 = (a,b,c)$ where $a,b,c \in {F_q}[T]$, in particular those such that the discriminant $D = b^2 - 4ac \in F_q[T]$ has even degree and sign ${D} \in ...
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0answers
20 views

finding the axis of a hyperbolic cylinder

I have data (a lot of points x,y,t) which are modeled by a hyperbolic cylinder $t^2 = b_0+b_1x+b_2y+b_3x^2+b_4xy+b_5y^2$ I know that if i just make a set of 6 equations from it, and than randomly ...
0
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1answer
25 views

determining a linear isomorphism so that two quadratic forms become equivalent

Consider the matrix $$ G = \begin{pmatrix} 3 & 1 & -2 \\ 1 & 2 & 0 \\ -2 & 0 & -3 \\ \end{pmatrix}$$ and the quadratic form $q: \mathbb{R}^3 \to \mathbb{R}$, given by $q(v) ...
0
votes
2answers
39 views

What does represent this equation?

$y^2 - 3z^2 + 4xz = 4$ Find its axis. As accurately as possible sketch its intersection with the plane $y = 0$. I tried with the making matrix showing the equation.
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3answers
83 views

How to sketch $-3x^2 - 8xy + 3y^2 = 1$ [closed]

The equation is as follows: $$-3x^2 - 8xy + 3y^2 = 1$$ How to specify the axis of the given curve? How to as accurately as possible draw a curve defined by this equation?
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1answer
29 views

Products of quadratic forms

It is known that, if $x_1^2 + y_1^2 = c_1$ and $x_2^2 + y_2^2 = c_2$, then $(x_1x_2 + y_1y_2)^2 + (x_1y_2 - x_2y_1)^2 = c_1 c_2$ Is there a similar analogue for general quadratic forms $Q(x, y) = ...
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0answers
19 views

isometric quadratic spaces over a prime field

Let $(V, \gamma)$ be a quadratic space, where $V$ is an $n$-dimensional $\mathbb{Z}/(7)$-vector space and $r = r(\gamma)$ is the rank of the bilinear form. I want to show: either, $(V, \gamma)$ is ...
0
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1answer
33 views

Variance of a quadratic form

I am considering a variance of two forms: $ R(x) = (x-m)^\top A (x-m) + b^\top (x-m) + c $ $ R'(\Delta) = \Delta^\top A \Delta + b^\top \Delta + c $ where $x$ is a random variable of ...
3
votes
1answer
49 views

Second order derivation of Quadratic form

I would like to find the second order derivative of a Quadratic form. Assume we have a random complex column vector $x$ and a real constant value $C$. I am interested in computing the following: $$ ...
4
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0answers
41 views

maximal linear subspaces contained in the cone over the Clifford torus.

Forgot: this is about Find a subspace of $\mathbb{R}^4$ for which $x^T*A*x$ = 0 I was a little surprised to find that, in the cone $x^2 + y^2 = z^2 + w^2$ in $\mathbb R^4,$ there are infinitely many ...
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4answers
145 views

How to solve an equation with $x^4$?

Today, I had this question on a Maths test about Algebra. This was the equation I had to solve: $$(1-x)(x-5)^3=x-1$$ I worked away the brackets and subtracted $x-1$ from both sides and was left with ...
3
votes
1answer
27 views

Why should the metrical groundform on a variety be a quadratic form?

I'm learning General Relativity and I can't understand why the distance function on space time is a quadratic form $$\textrm{d}s^2=g_{\mu\nu}\textrm{d}x^{\mu}\textrm{d}x^{\nu}$$ I explain it through ...
2
votes
1answer
40 views

Mass of a wire: intersection of surfaces

So I got this mass problem to solve: Find the mass of the wire formed by the intersection of two surfaces whose density is $\phi=x²$ $\underset{C}\int \phi ds $ along the curve: $$ C:\left\{ ...
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0answers
26 views

Quadric form $ax^2-ay^2$ to $x^2-y^2$

Let $\mathbb{F}$ be field with characteristic $\ne 2$. And $q = a(x^2-y^2)$ - quadric form on $\mathbb{F}^2$. I want to prove that there is some basic such such $q = x'^2 - y'^2$. I have proved this ...
0
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1answer
19 views

Simplifying an unusual quadratic linear algebra expression

I came across the following expression when solving a maximisation problem. I have the following ingredients: Matrices $\Omega, P \in \mathbb{R}^{n \times n}$ Vector $t \in \mathbb{R}^n$ Also let ...
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0answers
16 views

question about isotropic subspaces

if $V$ is a complex vector space of dimension $2n$ and $Q$ a bilinear form over $V$, the definition of an isotropic subspace is the following: $$\Lambda:Q(\Lambda,\Lambda) \equiv 0$$. Suppose that ...
0
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1answer
35 views

Finding the diagonal representation of a quadratic form

Let $q:\mathbb{R}^n\to\mathbb{R}$ be a quadratic form: $$q(x_1,\dots,x_n)=\sum_{i=1}^{n} x_i^2+\sum_{1\leq i < j \leq n} x_i x_j$$ I must find the diagonal form of $q$. My attempt: I tried ...
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0answers
13 views

$\text{SO}(2,1)$ is isomorphic to the group of automorphism of $x^2+y^2-z^2$

I am not able to prove the following. Let $\mathbb{K}$ be a field with more than four elements and with characteristic different from $2$. Let $\mathcal{C} \subset \mathbb{P}_2(\mathbb{K})$ be the ...
2
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2answers
61 views

Quadratic matrix equation: ellipse of all solutions

Consider the following equation in $Z$: $$-2 (\pmb X^T Y)^T Z+Z^T(\pmb X^T \pmb X)Z = 0$$ where: $\pmb X\in\mathbb{R}^{n\times p}$ and $Y\in\mathbb{R}^n$ with $n>p$ are known and ...
2
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2answers
36 views

Are reducible Integral Binary Quadratic Forms equivalent?

By an integral binary quadratic form (IBQF for short) I mean an $$f(x,y) = ax^2 + bxy + cy^2$$ with $a,b,c \in \mathbb{Z}$. Note that I am not assuming that they are all coprime. Such an $f$ is said ...
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2answers
54 views

Compute $f_A(\lambda)$ without factoring cubic polynomial?

I'm given the following prompt: "Find the points closest to the origin on the surface defined by $x_1^2+2x_2^2+3x_3^2+x_1x_2+2x_1x_3+3x_2x_3=1$." What's the easiest way to compute the ...
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0answers
15 views

Linear Algebra quadratic forms (max and plot)

If I have $q(x)=x_1^2-x_1x_2-x_1x_3+x_2x_3$ How do I find the maximum value of $q(x)$ subject to the constraint $||x||=4$? I already know the max when $||x||=1$ since it is the eigenvalue, but I don't ...
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1answer
47 views

Reduction of quadratic forms

To reduce a quadratic form $q: \mathbb R^n \longrightarrow \mathbb R$, one can: $1)$ Use the method of Gauss. For instance, if we have: $q: \mathbb R^3 \longrightarrow \mathbb R$: $q(x_1,x_2,x_3) = ...
30
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2answers
819 views

Meaning of the identity $\det(A+B)+\text{tr}(AB) = \det(A)+\det(B) + \text{tr}(A)\text{tr}(B)$ (in dimension $2$)

Throughout, $A$ and $B$ denote $n \times n$ matrices over $\mathbb{C}$. Everyone knows that the determinant is multiplicative, and the trace is additive (actually linear). \begin{align*} \det(AB) = ...
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0answers
29 views

How to graph quadratic forms and label points closest to and furthest from the origin?

$x_1^2+4x_2^2+9x_3^2=1$ $x_1^2+4x_2^2-9x_3^2=1$ $-x_1^2-4x_2^2+9x_3^2=1$ I have to sketch these three surfaces and determine which are "bounded", which are "connected", and what the points ...
2
votes
2answers
40 views

How can the level curves of a quadratic form be a pair of lines?

$x_1^2+4x_1x_2+4x_2^2=1\Rightarrow A\begin{pmatrix}1&2\\2&4\end{pmatrix}\Rightarrow ...
2
votes
1answer
38 views

Express a quadratic form as a sum of squares using Schur complements

So I was able to figure out the first part of this problem, but I have no concept of how it relates to Schur complements, so I'm not sure (no pun intended) how to proceed. The question is as follows: ...
0
votes
1answer
77 views

How do i expand/simplify this quadratic (or quartic?) equation

I'm having trouble doing the following question, was wondering if anyone was able to lend a hand, would be greatly appreciated as i'm not too sure where to start or how to go about this. The ...
1
vote
1answer
35 views

If equation has integer solution it has solution for every prime p.

How to prove that if the equation in the form: $a_0 x_0^2 + a_1 x_1^2 + \dots + a_nx_n^2 = 0$ where $a_0, a_1, \dots , a_n \in \mathbb{Z}$, has an integer solution, then it has solution in ...