Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [quadratic-forms]

A quadratic form is a homogeneous polynomial of degree two (in any number of variables), for example $4x_1^2 + 3x_1x_2 + 5x_1x_3 - 8x_3^2$.

1
vote
1answer
21 views

Find basis which fits quadratic form

Let $g(x,y)=ax^2+bxy+cy^2$ for some real $a,b,c$. Find a basis (or show one exists) $\{v_1,v_2\}$ for $\mathbb{R}^2$ such that $g(xv_1+yv_2)=gx^2+hy^2$ where $g,h=\pm 1$ or $0$. A hint would be ...
3
votes
2answers
36 views

How to show that a symmetric matrix that is strictly diagonal dominant (sdd) is is positive and definite?

Let $A\in M_n(\mathbb{R})$ be a real symmetric matrix that is strictly diagonal dominant (sdd) with positive diagonal values, that is, $$\forall i, a_{ii} = |a_{ii}|>\sum_{j\neq i}|a_{ij}|.$$ I ...
0
votes
0answers
11 views

Find out all equivalence classes?

Two Integer symmetric matrixes $A$ and $B$ are called equivalent with each other, if there exists $γ \in SL(2,Z)$ such that $B = γ^{T}Aγ$. Obviously, equivalent matrixes share the same determinant. ...
0
votes
0answers
9 views

Show that $\text{rank}(q)\leq \text{rank}(f_1,f_2,\cdots,f_p).$

Give a quadratic form $q = \sum_{i=1}^{p}\alpha_i f_i^2$ where $f_i \in E^{*}$ and $\alpha_i\in \mathbb{K},$ show that $\text{rank}(q)\leq \text{rank}(f_1,f_2,\cdots,f_p).$ I think this makes sense ...
0
votes
0answers
9 views

Determine all equivalence classes of a given quadratic form?

Let $G(x,y)$ be an integral coefficient binary quadratic form, which is positive definite. If the discriminant of $G(x,y)$ is $-56$, then I’m to determine all equivalence classes of $G(x,y)$. Say $G(...
6
votes
2answers
37 views

$p=a^2+ab+41b^2$ iff $-163$ is a quadratic residue

Prove that a prime $p$ can be written as $p=a^2+ab+41b^2$ iff $-163$ is a quadratic residue modulo $p$. What I have in mind is something like this: look at $\mathbb{Q}[\sqrt{-163}]$ which has the ...
0
votes
0answers
27 views

Serre’s definition of integral quadratic form

Chapter V of Serre’s book A Course in Arithmetic is titled ‘Integral Quadratic Forms with Discriminant $\pm1$.’ In the first section, he defines an object $E$ in a category $S_n$ (indexed by integers $...
0
votes
1answer
38 views

Simple quadratic inequality

Show that $${\bf x^T V x}\leq \left( \sum_{i=1}^n v_{ii}x_i \right)^2$$ knowing that $\bf V$ is symmetric and positive semidefinite. It should be simple, but can't figure it out. Thanks!
0
votes
0answers
24 views

Write a quadratic form as the sum of two non-degenerate quadratic forms.

Let $E$ be a finite dimensional $\mathbb{K}$ vector space and let $q : E \to \mathbb{K}$ be a quadratic form. Prove there exists two non-degenerate quadratic forms $q_1,q_2 : E \to \mathbb{K}$ such ...
1
vote
1answer
46 views

Transforming Quadrics in Characteristic 2

I’m trying to solve the following problem given in a textbook: Let $k$ be an algebraically closed field and $Q=V(F)$ a quadric in $\mathbb{P}^3(k)$, where $F$ is an irreducible polynomial in $X,Y,Z,...
1
vote
1answer
27 views

What is the procedure finding the Orthogonal Basis of a Quadratic Form?

We have the following quadratic form: $$q := 6 x^2_1 + 3 x^2_2 + 3 x^2_3 - 4 x_1 x_2 + 4 x_1 x_3 - 2 x_2 x_3$$ whose eigenvalues are $\lambda_1=\lambda_2=2$ and $\lambda_3=8$ I am not really sure ...
0
votes
1answer
44 views

Quadratic Form as Sum of Squares

I’ve been trying to prove the following: Let $k$ be an algebraically closed field with $\text{char}(k)\neq2$, and let $Q$ be a non-singular quadratic form on $k^n$. Show that for some choice of ...
0
votes
1answer
24 views

Bounds on representations via (non-positive) binary quadratic forms

Suppose that for some $n\in \mathbb{Z}$ we know that there are $x,y\in \mathbb{Z}$ such that $x^2-dy^2=n$ for some $d\in \mathbb{N}$. Can we say anything about how large $x$ and $y$ are compared to $n$...
1
vote
1answer
72 views

Squares in $\mathbb{Z}_p$

Consider the integral binary quadratic form $$f(x,y) = 2Axy+Bx^2$$ with $A,B \in \mathbb{Z}$ different from $0$. In Cassel's book "Rational quadratic forms" page 237 he claims that for $p \neq 2$ ...
0
votes
0answers
19 views

Calculating scalar product with given quadratic form

w, y, x $\in V$, compute <w,y> from given quadratic form $q(x) = $ $x^T Ax$ My approach is: $q(w)q(y) =$ $w^T Aw\cdot y^T Ay$ $=<w,y>w^T A^2 y$ I am wondering how to find $w^T A^2 y$. ...
0
votes
0answers
12 views

Quadratic forms which represent the same element

Let $a,b \in F^x$. Show that for quadratic forms holds: $D(\lt 1,a \gt) \cap D(\lt 1,b \gt) \subseteq D(\lt 1,-ab \gt)$ Here these sets represent the set of elements in $F^x$ which are represented ...
0
votes
0answers
42 views

Is the transpose of a matrix exponential with itself positive semi-definite under certain conditions

Given is a matrix $\boldsymbol{A}\in\mathbb{R}^{n\times n}$ with linearly independent eigenvectors $\boldsymbol{u}_1,...,\boldsymbol{u}_n$ and corresponding eigenvalues $\lambda_1,...,\lambda_n$. ...
0
votes
1answer
25 views

Notion of equivalence for intersection forms

Suppose $Q_X$ and $Q_Y$ are intersection forms of simply connected, smooth, closed 4-manifolds $X$ and $Y$. By Freedman, if $Q_X$ is equivalent to $Q_Y$, then $X$ is homeomorphic to $Y$ (though ...
1
vote
0answers
75 views

About the maximum distance between a point on a trajectory of a dynamical system, and its projection onto its linear interpolation

Summary This is a question regarding the maximum error that sampling of a dynamical system trajectory introduces w.r.t. the chosen time-step $\delta$. I formulate this as an optimization problem of ...
2
votes
0answers
23 views

Representability of primes by quadratic forms and congruence conditions

Let $$Q(x,y) = ax^2 + bxy + cy^2, \quad a,b,c \in \mathbb{Z}$$ be a binary quadratic form. We say an integer $n$ is representable by $Q$ if $n = Q(x,y)$ for some $x,y \in \mathbb{Z}$. A theorem due to ...
1
vote
0answers
19 views

primitive representation of integers over $\mathbb{Z}_p$

In Cassel's book "Rational quadratic forms" page 235 he claims that the form $$x_1^2 + x_2^2 +5(x_3^2 + x_4^2)$$ primitively represents $3 \cdot 2^{2m}$ over $\mathbb{Z}_p$ for every prime $p$ and ...
1
vote
1answer
16 views

Show that quadratic form is anisotropic over rational numbers

How to determine if quadratic form $q(x,y,z)=-6x^2+ \frac{7}{2}y^2-\frac{25}{7}z^2$ is anisotropic over $\mathbb{Q}$? This quadratic form is a diagonalisation of another quadratic form. Is there any ...
0
votes
0answers
33 views

A question about Quadratic forms

Assume $K$ is a field of characteristic different from $2$. Let $q=\langle a_1, \ldots, a_n \rangle$ be a quadratic form over a field $K$ such that $a_1, \ldots, a_n$ satisfy some algebraic relation, ...
0
votes
1answer
38 views

Write Lack of Fit Sum of Squares in Quadratic Form

Let \begin{equation} SSLF = \sum_{i=1}^{m}n_{i}(\bar{y_{i}} - \hat{y_{i}})^{2} \end{equation} then \begin{equation} \sum_{i=1}^{m}n_{i}(\bar{y_{i}} - \hat{y_{i}})^{2} = n(\bar{\overrightarrow{y}} -...
1
vote
2answers
23 views

How many solutions are for the quadratic equation $(12+a)x^2+12ax+9a=0$

I need to give a result for each $a\in \mathbb{R}$ how many solution is there to this equation: $(12+a)x^2+12ax+9a=0$. My attempt: Check for $B^2-4AC:$ $B^2-4AC = 144a^2-4(108a+9a^2)=108a^2-432a=...
0
votes
0answers
26 views

Why are quadratic forms required to be finite-dimensional?

Wikipedia defines a quadratic form over a field $K$ to be a map $q: V \to K$ from a finite-dimensional vector space over $K$ to $K$ such that $q(av) = a^2q(v)$ for all $a \in K, v \in V$ and the ...
0
votes
0answers
17 views

What does it mean for a Quadratic Form to be Weakly Positive Definite?

I am reading some lecture notes where it states that the difference of two metrics needs to be weakly positive definite. Since the metric is a symmetric bilinear form, I take it that this means that ...
1
vote
2answers
33 views

Reducing the quadratic form

I'm trying to reduce the quadratic form $q(x_1, x_2, x_3, x_4) = x_1x_2 + x_1x_3 + x_1x_4 + x_2x_4$ into a quadratic form of the form $q = λ_1y_1^2 + λ_2y_2^2 + ··· + λ_ry_r^2$ for some real numbers $...
0
votes
1answer
37 views

For what values of $b$ is this quadratic form indefinite?

I'm trying to determine what (real) values of $b$ the quadratic form $q = bx_1^2 + 2bx_2^2 + (9b + 2)x_3^2 − 2bx_1x_2 − 6bx_1x_3 + 4bx_2x_3$. I know (using leading principal of minors) that the ...
0
votes
1answer
16 views

Conditions when the system of symmetric quadratic equations has a solution

suppose we have a system of quadratic equations of the following form \begin{align} \boldsymbol x^T\boldsymbol A_i (\boldsymbol x + \boldsymbol b)=y_i\ ,\quad i=1,\cdots,M \end{align} where $\...
0
votes
0answers
65 views

Who was the first person to prove that only primes of the form $4k+1$ can evenly divide odd integers of the form $n^2+1$?

I am writing a paper and I want to cite the person(s) who proved that only primes of the form $4k+1$ can evenly divide odd integers of the form $n^2+1$. Edit: added "odd" For example, if $n=8$, then ...
0
votes
2answers
33 views

How to prove using the definition that matrix $A$ is negative definite?

I'm trying to prove that $$A=\begin{bmatrix} -2 & 2\\ 2 & -5\\ \end{bmatrix}$$ is negative definite. I can do it using the principal minors, but I am trying to understand how to ...
0
votes
1answer
53 views

Understanding Witt's Theorem

I just began to learn something about classical polar spaces and now, I'm trying to understand three implications of Witt's theorem. Let $V$ be an $m$-dimensional vector space over a field $K$ ...
1
vote
0answers
19 views

Definiteness of quadratic form subject to linear constraint

I have three questions related to the definition given in the image above: Why do we only have a restriction $q\neq0$? Shouldn't we place the same restriction on p as well? Is there a condition for a ...
0
votes
0answers
19 views

ternary quadratic form as a sum two squares of linear forms

Let $\phi(x,y)=ax^2+2bxy+cy^2$ be a binary quadratic form with integer coefficients. Pall proved in 1951 that $\phi(x,y)$ can be written as the sum of two squares of two linear forms if and only if $...
0
votes
2answers
34 views

Isotropic vectors of quadratic form

How can i find all isotropic vectors of quadratic form $x_{1}^2+4x_{2}^2+8x_{3}^2-4x_{1}x_{2}+8x_{1}x_{3}-14x_{2}x_{3}$ I don't understand how to approach.
2
votes
1answer
38 views

Is there a functional (i.e. infinite-dimensional) generalization of the second partial derivative test?

For a smooth function $f: \mathbb{R}^n \to \mathbb{R}$, we can (usually) test whether a critical point ${\bf x}_0$ (at which ${\bf \nabla} f({\bf x}_0) = {\bf 0}$) is a local maximum, minimum, or ...
0
votes
0answers
24 views

Quadratic forms + and -

Be Qº, Qºº: R2 ----> R1 two quadratic forms and the following statements: (I) If Qº and Qºº are positive definite then Qº + Qºº is positive definite. (II) There are Qº and Qºº definite positive such ...
1
vote
1answer
37 views

integrating an expression with alomst quadratic denominator including Chebyshev polynomials of the first kind

As a part of my research (in array processing - the specifics are not too exciting :-)), I cant resolve one specific integral. Assume $r\in\left[0 ,1\right]$, $N\in\mathbb{N}$. The basic form is $$...
0
votes
0answers
19 views

Unable to solve quadratic equation implemented and plotted in python

I model the inverter efficiency by eff = f(P_in) which is normed by P_in_pu = P_in/P_nomresulting in ...
0
votes
0answers
20 views

An inequality for a quadratic form and an inner product, and its relationship to the singular value decomposition

Let $A$ be a $p \times q$ real matrix of full column rank $q$, and let $u, v$ be two real vectors of (euclidean) norm 1. I want to know whether the following inequality holds: $$ v^TA^TAv \geq v^T A^...
3
votes
2answers
87 views

In what sense are Minkowski spaces with $(1,3)$ and $(3,1)$ signature isomorphic?

There is no isometry between a $(1,3)$-signatured and a $(3,1)$-signatured Minkowski space, but in spite of this, they "look like" the same, for example, they have the same light cone. Is there any ...
0
votes
2answers
22 views

Norm of a set of vectors with respect to a quadratic form

I've got a problem that I'm struggling to put into a form that I can analyze. Suppose I have a quadratic form $f(x,y)=ax^2+2bxy+cy^2 = \mathbf{u}\mathbf{A}\mathbf{u}^T$ for $\mathbf{u} = \begin{...
0
votes
1answer
45 views

Positive definite quadratic form over positive integer vectors

Let $\mathcal{M}$ denote the set of $d \times d$ positive definite matrices with real entries for some positive integer $d$. Now consider the set $$ \mathcal{N}=\{A \in M_n(\mathbb{R}): z^TAz > 0, ...
0
votes
0answers
23 views

Unique multiplicative quadratic form on quaternion algebras

I want to prove, that the only multiplicative quadratic form $Q$ (so $Q(xy)=Q(x)Q(y) \forall x,y$) on a quaternion algebra $\Big(\dfrac{a,b}{F}\Big)$ is the norm $\mathrm{Nr}$, which is isometric to ...
2
votes
1answer
56 views

on the quadratic form $5x^2-y^2$

Consider the following subsets of $\mathbb{Z}$: $A=\{\frac{5p^2-y^2}{4}\, | \, p, y \,\,\text{odd positive integers}, p\,\, \text{prime}\}$ and $B=\{\frac{5x^2-y^2}{4}\, | \, x, y \,\,\text{odd ...
0
votes
0answers
32 views

Nonlinear Recursion Solution Process for $x_{n+1}=\Sigma_{i=1}^{n} x_{i}x_{n-i}$ (Known Solution)

I want to solve the equation $x_{n+1}=\Sigma_{i=1}^{n} x_{i}x_{n-i}$. Plugging the equation into Mathematica gives me $x_n=(-1)^{n}2^{2n+1} Binomial(1/2, n+1)x_0^{n+1}$. How might I derive this?
0
votes
1answer
24 views

Creating a quadratic equation from a condition

how can I create an equation that satisfies the following: "x-intercepts 1 and -1, y-intercept 3". I understand that the factored form of a quadratic equation offers both x intercepts however, I'm not ...
3
votes
5answers
44 views

The square trinomial $y=ax^2+ bx + c$ has no roots and $a + b + c > 0$. Find the sign of the coefficient $c$ .

The square trinomial $y=ax^2 + bx + c$ has no roots and $a + b + c > 0$. Find the sign of the coefficient $c$. I'm having difficulties with this problem. What I've tried: I realized that a ...
1
vote
1answer
27 views

Orbits of the conjugation action of $GL_3 (\mathbb{R})$ on the nonsingular symmetric $3\times3$-matrices

Let $S$ be the space of all symmetric $3 \times 3$ matrices of full rank and with real entries. $GL_3 (\mathbb{R})$ acts on this space by conjugation, \begin{align*} g.A = (g^{-1})^T A g^{-1}, \quad ...