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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0
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2answers
19 views

Solve the algebraic expression for a, b, and c of the function x

I am trying to solve for $a$, $b$, and $c$ in the expression below, but I have found that the way I tried to solve it is convoluted and did not work out. I believed that by solving for x, I would be ...
5
votes
2answers
87 views

How are the pseudo-Riemannian metric tensor properties restricted by the manifold topology in pseudo-Riemannian manifolds?

My understanding is that a pseudo-Riemannian metric tensor induces a topology that is not compatible with the manifold topology, and obviously the manifold topology prevails if we are to have a ...
0
votes
0answers
11 views

Reducing a pair of indefinite quadratic forms to the canonical form

Assume $A, B$ being a pair of symmetric matrices over reals. Let $$ \varphi_1(x) = (x, Ax)\\ \varphi_2(x) = (x, Bx). $$ There's a well-known result that if $A > 0$ then the pair of forms can be ...
5
votes
0answers
43 views

Simultaneously vanishing quadratic forms?

Given a set of Hermitian matrices $\{A_i\}$, is there a simple way to check if there exists a vector $c$ such that for all $i$: $$c^* A_i c = 0?$$ Namely, when can the quadratic forms defined by the ...
1
vote
1answer
29 views

Polarity on a Hyperboloid of one sheet

Given a quadric $Q = \{v \in \mathbb{R}^n \mid \alpha(v,v) = 1\} \subset \mathbb{R}^n$, defined by a bilinear form $\alpha: \mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}^n$, and an affine subspace ...
0
votes
0answers
151 views

Proof of Fisher-Cochran's theorem

Let $dim(E)=n$ We have $u_1, u_2..., u_p$ self-adjoint operators which belong to $E$ $(i)$ : $rk(u_1)+...+rk(u_p)=n$ $(ii)$ : $q_1(x)+...q_p(x)=x.x$ with $q_i$ the quadratic form $q_i(x)=u_i(x).x$ ...
1
vote
1answer
33 views

Quadratic form as generalized distance?

In the book A Linear Systems Primer (by Antsaklis and others), they first mention squared distance of a point x from the origin: $$x^{T}x = ||{x}||^2$$ which represents the square of the ...
0
votes
2answers
43 views

Quadratic Forms and Associated Matrices

This might be a dumb question but when we write the matrix associated with a quadratic form, why does the matrix need to be symmetric in general? I'm asking because I'm thinking there isn't a unique ...
-1
votes
2answers
43 views

Can this expression be made into a quadratic form?

Can this expression be made into a quadratic form: $ a x_t -\gamma {x_t}^2 $ I want to solve a linear quadratic programming problem and it requires that I put this expression in a quadratic form. $ ...
0
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0answers
9 views

How to do linear quadratic dynamic programming with non homogeneous quadratic equation

I am not well versed on matrix algebra and linear quadratic programming. I am wondering if it is possible to make a non-homogeneous equation into a homogeneous one. I need to make the following ...
2
votes
2answers
6k views

Derivative of Quadratic Form

For the Quadratic Form $X^TAX; X\in\mathbb{R}^n, A\in\mathbb{R}^{n \times n}$ (which simplifies to $\Sigma_{i=0}^n\Sigma_{j=0}^nA_{ij}x_ix_j$), I tried to take the derivative wrt. X ($\Delta_X X^TAX$) ...
0
votes
1answer
16 views

Almost universal integer quadratic forms

This question is inspired by the 15-theorem. For any nonnegative integer k, define a k-universal integer quadratic form to be a form that represents all but k positive integers. So, universal forms ...
0
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0answers
41 views

Question about Mumford's article

I'm reading the following article by Mumford speaking about theta characteristic. Mumford's article I'm trying to understand the definition af the quadric form $q$ on page 184. Here my questions: 1) ...
-1
votes
0answers
48 views

Solving equations of degree 2

Is there a program that can solve equations of degree 2? For example, if we input a system of n equations with left sides being quadratic forms in n varialbes i.e. of the form ...
0
votes
0answers
27 views

how to determine a biquadratic form is positive-definite

A biquadratic form $\sum_{i,j,k,l}b_{i,j,k,l}x_{i}x_{j}y_{k}y_{l}$, how to determine whether it is positive-definite? A necessary and sufficient condition? In fact, I have a matrix $B=\sum_{1\leq ...
3
votes
0answers
28 views

Quadratic form and matrix

We know quadratic form $f(x_1,x_2)= a_{11} x_1^2 + 2 a_{12} x_1 x_2 + a_{22} x_2^2$ is non-negative for all $x_1,x_2 \in \mathbb{R}$ iff matrix $(a_{ij})_{2 \times 2}$ is semi-positive defined. My ...
0
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0answers
25 views

Complexity of computing a posiform of a quadratic pseudo-boolean function

I am reading the chapter 13, Pseudo-Boolean functions, of Boolean Functions: Theory, Algorithms, and Applications by Crama et. al. In section 13.2, the authors introduce the idea of Posiform. The ...
0
votes
1answer
34 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 - ...
-1
votes
2answers
19 views

graph quadratic form and find the equation of asymptotes

So I had this quadratic form that need to be graphed showing both original and new axes. And I also need to find out the equation of asymptotes. $$ \left\{ \begin{aligned} ...
0
votes
1answer
20 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 $|x_i|\leq 1$, ...
0
votes
2answers
36 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 ...
1
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0answers
27 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 ...
2
votes
2answers
79 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 ...
0
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0answers
29 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 ...
1
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0answers
15 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 ...
0
votes
1answer
14 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 $a,b \in \mathbb{R}.$ ...
4
votes
1answer
52 views

Pell's equation and representation elements of $\mathbb Z_p$.

We defined the function $f:\mathbb Z_p \times \mathbb Z_p \rightarrow \mathbb Z_p$ such that $f(x,y)=x^2-cy^2$ and $c\not\equiv 0\pmod{p}$. Is it true that $f$ is onto?
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0answers
9 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 ...
0
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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) ...
2
votes
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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vote
3answers
132 views

Solving Series of equations

I have the following series of equations ($n+2$ equations $n+2$ variables): \begin{equation*} k_0q_0+\lambda q_0 + c_0 = 0, \\ k_1q_1+\lambda q_1 + c_1 = 0, \\ k_nq_n+\lambda q_n + c_n = 0, \\ ...
0
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0answers
9 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 ...
0
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0answers
8 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 ...
1
vote
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 ...
0
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0answers
32 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
votes
1answer
15 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
votes
2answers
32 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 ...
3
votes
2answers
7k views

Find the EigenValues and EigenVectors of the matrix associated with quadratic form

Problem: Find the EigenValues and EigenVectors of the matrix associated with quadratic forms $2x^2+6y^2+2z^2+8xz$. I know how to convert a set of polynomial equations to a matrix but I have no clue ...
8
votes
1answer
83 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 ...
0
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0answers
11 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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0answers
51 views

Show that for any given $d<0$, the primitive positive definite quadratic forms of discriminant $d$ all have the same number of automorphs.

Show that for any given $d<0$, the primitive positive definite quadratic forms of discriminant $d$ all have the same number of automorphs. I think we should let $f(x,y) = ax^{^{2}} + bxy + ...
0
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0answers
17 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 $$u^t V u + x^t V x + u^t V x + x^t V u$$ ...
1
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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 ...
0
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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 ...
0
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0answers
21 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 ...
3
votes
1answer
50 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: $$ ...
0
votes
1answer
27 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
41 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.
0
votes
3answers
88 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?
0
votes
1answer
31 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) = ...