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

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2
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3answers
98 views

Integer solutions to $x^2-xy+y^2=1$

What are the integer solutions to $x^2-xy+y^2=1$? (I found the solution below while working on another problem, so I thought I'll add it to the knowledge base here.)
1
vote
0answers
58 views

Pell's equation and binary hyperbolic forms.

We define the function $f:\mathbb Z_p \times \mathbb Z_p \rightarrow \mathbb Z_p$ such that $f(x,y)=x^2-cy^2$ and $0\neq c=a^2+b^2$. Is it true that $f$ is hyperbolic? In other word, is there any ...
0
votes
0answers
14 views

Quadratic forms with the same discriminant

Show that any quadratic form $ax^2+bxy+cy^2$ with discriminant $-4$ is equivalent to the form $x^2+y^2$, and any quadratic form of discriminant $-3$ is equivalent to $x^2+xy+y^2$. Here quadratic ...
0
votes
0answers
14 views

Approximating a quadratic term in the constraint set as 2nd order Taylor expansion

I have an optimization problem in the following form: $$\min_{x,y} f(x)+g(y)$$ $$s.t.$$ $$Ax+h(y)=0$$ where $h(y)$ is a quadratic in $y$. Instead of solving this problem directly, $h(y)$ is ...
2
votes
2answers
1k views

diagonalize quadratic form

I have this quadratic form $Q= x^2 + 4y^2 + 9z^2 + 4xy + 6xz+ 12yz$ And they ask me: for which values of $x,y$ and $z$ is $Q=0$? and I have to diagonalize also the quadratic form. I calculated ...
0
votes
0answers
18 views

Sesquilinear Forms

I was trying to solve some exercises related to sesquilinear forms: Let V be a C-vector space (C - complex numbers) Prove that the set $\mathcal{S}(V)$ of sesquilinear forms on V is a vector ...
0
votes
0answers
21 views

Maximization of quadratic form on a sphere [duplicate]

I have to following problem $$\max_{x}x^TAx+b^Tx\quad \mathrm{s.t.}\quad x^Tx\leq c,$$ where $A$ is real, symmetric and positive semi-definite. Firstly I tried to solve the problem with the KKT, but ...
2
votes
2answers
192 views

Calculus approach to solve this Quadratic equation problem

Both roots of the equation $$(x-b) (x-c) +(x-a) (x-c) +(x-a) (x-b) = 0$$ are always positive , negative or real. Prove your result. By solving this equation I got $3x^2 - 2(a+b+c)x +ab + bc + ca ...
0
votes
1answer
37 views

Show that if a quadratic form is primitive then so are equivalent forms

A Quadratic form is primitive if the greatest common divisor of the coefficients of it's terms is 1. I saw in number theory book that "it is easily seen that any form equivalent to a primitive form ...
13
votes
1answer
2k views

Proof of Legendre's theorem on the ternary quadratic form

Theorem (Legendre): Let $a,b,c$ coprime positive integers, then $ax^2 + by^2 = cz^2$ has a nontrivial solution in rationals $x,y,z$ iff ...
0
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0answers
26 views

what is a ordinally quadratic function?

A function is ordinal equivalent to another means there exist a (unique) monotonic transformation between wiki definition of ordinal utility. I am a little confused, a function is ordinally quadratic ...
0
votes
0answers
10 views

Maximize function symbolically

I have the following expression: $$ \sum_{i,j=1}^n\rho_{ij}^2-\frac{2}{n}\sum_{i=1}^n\left(\sum_{j=1}^n\rho_{ij}\right)^2 +\frac{1}{n^2}\left(\sum_{i,j=1}^n\rho_{ij}\right)^2 $$ My goal is to ...
7
votes
2answers
112 views

Solving a quadratic 9-equation system

I need to solve the following system: $$\begin{cases} A^TA=B &(1)\\ A\vec{x}=\vec{y} &(2)\\ \end{cases} $$ I need $A$, given $B$, $\vec{x}$ and $\vec{y}$. $A$ and $B$ are both 3-by-3 ...
0
votes
2answers
43 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
0answers
18 views

Completing a multivariate square

A well-known trick when analyzing quadratic polynomials $P=ax^2+bx+c$ is to complete the square: P can be written as $$P=\left(x+\frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a^2}.$$ I have been trying to ...
1
vote
1answer
60 views

Transforming Diophantine quadratic equation to Pell's equation

I have been discussing the fastest and most efficient ways of solving QDEs in a separate question record (Alternative method to solve quadratic Diophantine equations). However, as suggested by ...
0
votes
1answer
13 views

Factoring binary quadratic form in two second order polynomials

I have a binary quadratic form in $N$ and $D$, $AD^2 + BND + CN^2$, where $A$, $B$, and $C$ are real coefficients and $N$ is a second order polynomial of $x$ with real roots $\lvert r \rvert <1$ ...
-1
votes
1answer
20 views

Quadratic forms matrices

Let $$Q(x,y,z) = – 2x^2 + 6xy + 8y^2 + z^2.$$ Find the symmetric matrix associated with this quadratic form. Use the determinant method to determine whether the quadratic form is positive definite, ...
2
votes
0answers
51 views

On the integer solutions to $u^2+163v^2=w^3$ and others

It seems the solution of, $$u^2+dv^2 = w^3\tag1$$ involves the class number $h(d)$. Assume $\gcd(u,v)=1$. Q: For which $\color{red}{prime}\; d$ is the complete solution of $(1)$ in the integers ...
6
votes
0answers
38 views

Why do isotropic spaces deserve their name?

Wiki defines a quadratic form to be isotropic if it evaluates to zero at some vector. What does this have to do with isotropy in physics i.e uniformity in all directions? From my experience so far, ...
1
vote
1answer
61 views

Maximization of vector norm under a quadratic convex inequality constraint

I need help for the following problem: $$ \max_x x^Tx\quad \mathrm{s.t.}\quad x^TAx+b^Tx\leq c, $$ where A is symmetric, square and positive semidefinite, c is a real scalar and b is a real vector. ...
4
votes
2answers
97 views

Unable to find solution for $a^2+b^2-ab$, given $a^2+b^2-ab$ is a prime number of form $3x+1$

I have a list of prime numbers which can be expressed in the form of $3x+1$. One such prime of form $3x+1$ satisfies the expression: $a^2+b^2-ab$. Now I am having list of prime numbers of form $3x+1$ ...
7
votes
3answers
62 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
1answer
57 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
26 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
55 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
1answer
72 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 ...
2
votes
1answer
30 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 ...
3
votes
0answers
34 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 $\phi_{IF}$ ...
0
votes
1answer
31 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
1answer
17 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. ...
1
vote
4answers
612 views

How can I find integer values for which a given expression gives a perfect square?

Find the integer values for which $x^2+19x+92$ is a perfect square. Also, How to proceed if you have to find values ( not necessarily integer)?
2
votes
1answer
63 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) &= ...
0
votes
0answers
10 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 $x^T A x$ ...
1
vote
1answer
41 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 ...
2
votes
1answer
29 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 ...
0
votes
1answer
53 views

Quadratic reciprocity and Pfister forms

Let $p,q$ be different primes unequal to $2$. Let $(a/b)$ denote the Legendre Symbol. The following holds: $q\text{ is a square }\bmod p \Longleftrightarrow (q/p) = 1 \Longleftrightarrow X^2+qY^2 = ...
4
votes
1answer
278 views

A characterization of an ambiguous class of binary quadratic forms of discriminant $D$

We use the definitions of this question. Let $D$ be a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). There exists a bijection $\psi\colon Cl^+(R) \rightarrow C(D)$ by ...
0
votes
0answers
20 views

Polarization of a quadratic Hermitian form

Suppose $f$ is a conjugate-symmetric sesquilinear form over a vector space $V$ defined over $F=\mathbb{F}_{q^2}$, i.e. $f:V\times V \rightarrow F$ such that $$f(\lambda u + v, w) = \lambda f(u, w) + ...
1
vote
1answer
40 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 ...
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votes
0answers
9 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^+ ...
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vote
0answers
31 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
42 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 ...
2
votes
1answer
171 views

Simultaneous Orthogonalization

Let $q,q':\mathbb V \longrightarrow \mathbb R$ be two quadratic forms, where $\mathbb V$ is vector space with $\dim \mathbb V \geq3$ and $q(x)+q'(x)>0$ for any $0\neq x\in \mathbb V$. Then there ...
2
votes
1answer
190 views

Binary quadratic forms - Equivalence and repressentation of integers

If $f,g$ are two binary quadratic forms, $f$ and $g$ are equivalent, if there is an integer matrix $M$ with determinant $\pm1$ such that $G=M^T F M$ where $F,G$ are the matrices that define $f,g$. It ...
0
votes
0answers
11 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
86 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 ...
4
votes
4answers
81 views

What are all the concordant forms $n$ such that $a^2+b^2 = c^2,\,a^2+nb^2=d^2$ for $n<1000$?

Part I. The list of congruent numbers $n<10^4$ such that the system, $$a^2-nb^2 = c^2$$ $$a^2+nb^2 = d^2$$ has a solution in the positive integers is known (A003273) $$n = 5, 6, 7, 13, 14, 15, ...
6
votes
1answer
278 views

Follow up on intersection forms

For which topological spaces $X$ can I define an intersection form $b(\cdot, \cdot)$? I know at least one example: If $X$ is a closed orientable $2n$-manifold then one can define an intersection ...