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

0
votes
1answer
60 views

Regarding the factorization $a^2+3b^2 = cd$.

Let $a,b,c,d$ be positive integers, with $\gcd(c,d)=1$, such that $$a^2+3b^2=cd.$$ By well-known classical results, we have that $c$ and $d$ are both of the form $u^2+3v^2$. QUESTION: Is it valid to ...
0
votes
1answer
31 views

Lower boundary of quadratic form

I have a quadratic form $x^TAx$ where $x$ is an $n \times 1$ vector and $A$ is a positive definite matrix in the sense that it has only positive eigenvalues. Am I right to say that $||x^TAx|| \ge \...
0
votes
1answer
56 views

Positiveness of a specified quadratic form

The condition of the positiveness of a ordinary quadratic form can be derived by getting the condition of positiveness of a square matrix, like ${v}^{T}{A}{v} \geq 0$ is equal to matrix $A \geq 0$ ...
3
votes
0answers
36 views

Statement about composition of binary quadratic forms in “A Course in Computational Algebraic Number Theory”

On p.239 A Course in Computational Number Theory, Cohen writes "Although the group structure on ideal classes carries over only to classes of quadratic forms via the maps $\phi_{FI}$ and $\phi_{IF}$ ...
2
votes
1answer
33 views

Finding the spinor norm of an element using a proposition in 'The Maximal Subgroups of the Low-Dimensional Finite Classical Groups'.

Let $F=\mathbb{F}_{q}$, where $q$ is an odd prime power. Let $e,f,d$ be a standard basis for the $3$-dimensional orthogonal space $V$, i.e. $(e,e)=(f,f)=(e,d)=(f,d)$ and $(e,f)=(d,d)=1$. I have an ...
7
votes
3answers
69 views

Fermat's Challenge of composition of numbers

In his letter to Carcavi (August 1659), Fermat mentions the following challenge There is no number, one less than a multiple of $3$, composed of a square and the triple of another square. ...
0
votes
2answers
45 views

Representations of some primes as $3x^2-4y^2$?

I am looking for (elementary) proofs (idea of the proofs is also OK) or references to proofs of the followings: $$ p\equiv11\pmod{12}\longrightarrow p=3x^2-4y^2 $$ Any help appreciated.
0
votes
1answer
20 views

Hurwitz's matrix equations

I have a question about the proof of Hurwitz's 1-2-4-8 theorem about the sum of squares. I have consulted Chapter 1 of Rajwade's "Squares" book, notes by Keith Conrad, and notes by Daniel Shapiro. ...
3
votes
1answer
73 views

How to solve similar with Transformation of a quadratic form into diagonal form?

Define $\color{red}{f=f(x),f'=f'(x)}$,where the derivative with respect to $x$ of a function $f(x)$ is denoted $ f'(x)$. Now give six postive numbers $k_{1},k_{2},k_{3},k_{4},k_{5},k_{6}$, and a ...
0
votes
1answer
32 views

Representations of some primes as $x^2-2y^2$?

I am looking for (elementary) proofs (idea of the proofs is also OK) or references to proofs of the followings: $$ p\equiv\pm1(\mod8)\longrightarrow p=x^2-2y^2 $$ Any help appreciated.
0
votes
0answers
11 views

Lagrangian of quadratic form with linear constraints

I am working through this paper, and I am confused as to how the author obtains the Lagrangian in equation (1.4) More specifically, in this paper, the authors present a problem: Minimize $x^T A x$ ...
2
votes
1answer
52 views

Quadratic form.

Consider the quadratic form $Q(v)=v^{t}Av,v=(x,y,z,w)$ where matrix $A$ is given by \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & \\ 0 & 0 & 0 &...
6
votes
1answer
120 views

On a remarkable system of fourth powers using $x^4+y^4+(x+y)^4=2z^4$

The problem is to find four integers $a,b,c,d$ such that, $$a^4+b^4+(a+b)^4=2{x_1}^4\\a^4+c^4+(a+c)^4=2{x_2}^4\\a^4+d^4+(a+d)^4=2{\color{blue}{x_3}}^4\\b^4+c^4+(b+c)^4=2{x_4}^4\\b^4+d^4+(b+d)^4=2{x_5}...
2
votes
1answer
30 views

Relation between Pfister forms in W(Q)

For $n$ a nonzero integer, is it possible to have a relation $$ n\langle\langle-1,-1\rangle\rangle=\sum_i n_i\langle\langle a_i,b_i\rangle\rangle$$ in $I^2(\mathbb Q)\subset W(\mathbb Q)$ (or maybe $...
2
votes
1answer
67 views

How canonical is Gauss's law of composition of forms

Gauss defined the composition of binary quadratic forms $f$ and $g$ to be another binary quadratic form $F$ such that there exist integral quadratic forms $$ \begin{align} r(x_0,x_1,y_0,y_1) &= ...
1
vote
1answer
47 views

How to write a given element of the orthogonal group as a product of reflections

Let $V$ be a 3-dimensional vector space over a finite field $F$ of $q$ elements, where $q$ is an odd prime power. We know that the orthogonal group $O(V)$ is generated by reflections. How can a given ...
0
votes
0answers
11 views

Maximizing $y^H ( I - X pinv(X) ) y $ with respect to matrix $X$. How hard can it get?

Assume $X$ to be a tall block-diagonal matrix where each block is a collumn vector. Assuming $X^+ = (X^H X)^{-1}X^H $ to be the pseudoinverse of the matrix $X$, find $X$ which maximizes $$y^H ( X X^+ ...
1
vote
0answers
34 views

Clifford Algebra Isomorphic to Exterior Algebra

Let $E$ be a vector space over a field $k$ and $Q$ be a quadratic form, that is, $$Q:E\to k$$ such that $$Q(\lambda e)=\lambda^2Q(e)\forall\lambda\in k\,e\in E$$ and such that $P_Q:E^2\to k$ is ...
3
votes
0answers
43 views

Integral of a multivariate Gaussian distribution over quadratically separated partions

Imagine in the space of $\Re^n$, the quadratic curve $c: f(\mathbf{x}) = \mathbf{x}^TW\mathbf{x} + \mathbf{w}^T\mathbf{x} + w_0$ (with $W$ being a symmetric positive definite matrix, $\mathbf{w}$ a ...
0
votes
0answers
12 views

Expectation of quadratic form of correlated variables

Suppose $U$ and $V$ are two $n\times 1$ dependent vectors, in the sense that $E\left( UV^{\prime}\right) \neq \mathbf{0}.$ For a given constant $n\times n$ matrix $A,$ is there is any simple way to ...
4
votes
1answer
97 views

What are numbers $n$ such that $a^2+nb^2 = c^2$ and $na^2+b^2 = d^2$?

Let $n$ and $a,b,c,d,$ be in the positive integers. I. For the system, $$a^2-nb^2 = c^2\\a^2+nb^2=d^2$$ then $n$ is a congruent number. The sequence starts as $n=5,6,7,13,14,15,20,21,$ and so ...
1
vote
0answers
53 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 the ...
0
votes
1answer
21 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 ...
0
votes
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 ...
0
votes
2answers
30 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 ...
0
votes
0answers
17 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 ...
0
votes
0answers
17 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 }, \prod_{i=1}^{k}{(v_{...
0
votes
1answer
43 views

Degenerate quadratic form [closed]

I'm beginning with quadratics forms and I am wondering : Let $a$ be a real number and $q:\mathbb{R^4} \rightarrow \mathbb{R}$ given by $$(x,y,z,t) \rightarrow ax^2+2axy+y^2+4zt-at^2.$$ I would ...
0
votes
2answers
34 views

Maximums on Quadratic Functions [closed]

How do you find the maximum of a quadratic function? Specifically, $R(x) = -4x^2 + 4000x$
0
votes
1answer
32 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 ...
1
vote
3answers
36 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 $p$ ...
1
vote
1answer
56 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 ...
1
vote
1answer
34 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 ...
2
votes
0answers
51 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 $\phi \colon R \times R$ ...
1
vote
0answers
47 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 ...
1
vote
0answers
30 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 + ...
2
votes
0answers
27 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 ...
1
vote
1answer
14 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} ...
0
votes
2answers
98 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 ...
0
votes
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 ...
2
votes
0answers
44 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 $\mathbf{u}^T\mathbf{...
2
votes
1answer
53 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?
2
votes
0answers
38 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 ...
0
votes
1answer
14 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
votes
1answer
41 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$?
2
votes
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 $1 \times 1$...
1
vote
2answers
65 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 ...
0
votes
1answer
27 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) ≤ ...
4
votes
0answers
49 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. ...
1
vote
1answer
67 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 form)...