Questions about quadratic forms in multiple variables, for example $4x_1^2 + 3x_1x_2 + 5x_1x_3 - 8x_3^2$.

learn more… | top users | synonyms

1
vote
1answer
51 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 $L ...
2
votes
1answer
164 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
52 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
votes
0answers
54 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 ...
0
votes
1answer
25 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
votes
0answers
42 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) ...
0
votes
0answers
39 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 i,...
3
votes
0answers
49 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
votes
0answers
52 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 ...
-1
votes
2answers
30 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} 4(x_1)^2-12(x_1)(x_2)-(...
0
votes
1answer
48 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 $|x_i|\leq 1$, $i=1,\dots,...
1
vote
0answers
45 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 ...
0
votes
1answer
42 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, $m=m_1 - \lambda*m_2$...
0
votes
2answers
51 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 ...
3
votes
1answer
114 views

Combining sums and/or differences of squares

I'd like to combine a sum of as many squares as possible into a sum of as few squares as possible. The signs of the squares doesn't matter. For example, the Brahmagupta-Fibonacci Identity combines a ...
1
vote
0answers
38 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 $$Q=...
1
vote
0answers
30 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
votes
2answers
100 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 ...
1
vote
0answers
32 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?
1
vote
1answer
38 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 is ...
0
votes
0answers
50 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 $n\...
0
votes
1answer
36 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
35 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
votes
1answer
100 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 symbol ...
0
votes
1answer
17 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 $a,b \in \mathbb{R}.$ We ...
0
votes
0answers
25 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
vote
0answers
39 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 \mathbb{Q}_2[\zeta+...
0
votes
0answers
29 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
votes
1answer
39 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
3answers
105 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?
1
vote
1answer
62 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) = ax^...
1
vote
0answers
27 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
votes
1answer
420 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 $\mathcal{N}(...
3
votes
1answer
73 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
votes
0answers
51 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 ...
5
votes
4answers
191 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
39 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
55 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\{ \...
0
votes
1answer
20 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 $\...
0
votes
0answers
56 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
votes
1answer
84 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 ...
1
vote
0answers
21 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
votes
2answers
102 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 $Z\in\mathbb{...
2
votes
2answers
47 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 ...
0
votes
2answers
59 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 ...
1
vote
0answers
17 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 ...
0
votes
1answer
159 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) = ...
32
votes
2answers
1k 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) = \...
1
vote
0answers
63 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 closest ...
2
votes
2answers
94 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 f_A(A)=\begin{vmatrix}1-\lambda&2\\2&4-\lambda\end{vmatrix}\Rightarrow\begin{Bmatrix}\lambda_1=0\\\...