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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1answer
38 views

Find $a^2 + b^2+c^2$

Given $a^2+2b = 7$ $b^2+4c = -7$ $c^2+6a = -14$ Find $a^2 + b^2 + c^2$ The answer was an Integer I tried to solve it by making $a$ the subject of the equation and substituting in others but ...
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2answers
99 views

About integral binary quadratic forms fixed by $\operatorname{GL_2(\mathbb Z)}$ matrices of order $3$

I am reading this paper of Manjul Bhargava and Ariel Shnidman, and I want to prove this claim, which appear at the first paragraph of Theorem $14$: Up to $\operatorname{SL_2}(\mathbb Z)$ ...
0
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1answer
48 views

A question on a nonnegative quadratic form

Denote $x,y,z$ as variables, and $a,b,c$ as coefficients. Suppose $a\leq b\leq 0\leq c$ and $a+b+c=0$. Could anyone help me prove whether the following quadratic form positive semi-definite? ...
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1answer
77 views

Non-standard quadratic matrix equation

I have an equation that looks like the following: $$ A\cdot\mathrm{diag}(x)\cdot x + B\cdot x + c = 0 $$ where $A, B, C \in \mathbb{R}^{n \times n}$ and $x, c \in \mathbb{R}^n$. $ x $ is unknown. ...
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1answer
149 views

A problem on positive semi-definite quadratic forms/matrices

Suppose $a+b+c=0$ and (without loss of generality) $a\leq b\leq 0\leq c$, $a^2+b^2+c^2=1$, is the following quadratic form positive semi-definite? Thank you very much. \begin{equation*} \begin{split} ...
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1answer
32 views

Scaling variables in homogeneous equation of degree two in a,b,c

The problem I'm having trouble with is: Let $a,b,c$ be nonzero real numbers and let $a^2 - b^2 = bc$ and $b^2 - c^2 = ca$. Prove that $a^2 - c^2 = ab$. The solution strategy given in the course was ...
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1answer
205 views

Equivalence for binary quadratic forms with positive square discriminant

I recently encountered an interesting proposition without proof: If $f(x,y)$ is a quadratic form whose discriminant is a non-zero perfect square, then $f(x,y)$ is equivalent to a form $a*x^{2} + ...
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0answers
82 views

Freeman Dyson's identity for the modular discriminant $\Delta$

In his 'Introduction to Modular Forms', Don Zagier states the Freeman Dyson's identity : $$\Delta(\tau)=\sum_{\substack{ (x_1,\ldots,x_5)\in \mathbb{Z}^5 \\ x_1+\cdots+x_5=0 \\ x_i \equiv i ...
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1answer
36 views

Quadratic equation form

I have the relation $u=\sqrt{(a_1+b_1t)^2+(a_2+b_2t)^2+(a_3+b_3t)^2} \tag 1$ I need to write $t$ as a function of $u$ ($t=f(u)$). How will I get that ? NB: $a_1,a_2,a_3,b_1,b_2,b_3$ are ...
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1answer
61 views

Discriminant of a Quadratic form

Let $V$ be a vector space over field $K$ and $Q$ is the quadratic form on it, and $A$ be the matrix w.r.t. $e_1,e_2,...e_n$ of $V$. Now $discr(Q)$ is defined as $det(A)$ mod ${K^{*}}^{2}$. Now my ...
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1answer
680 views

How to show that two quadratic forms are equivalent?

To show to quadratic forms are not equivalent, we can find rank, or discriminant or some element which is represented by either one only etc. But Is there a general criterion to show that two ...
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2answers
37 views

Solve a system of equations when one is linear and the other is quadratic

$x+y=3m$ $xy=2m^2$, $m$ is the parameter. I came to this $2m^2-3mx+xy=0$. The solutions have to be:$(m,2m),(2m,m)$. But I can't understand what is the role of this parameters, I don't know how to ...
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3answers
36 views

System of two equations with two unknowns - can't get rid of $xy$

The system is: $x^2 + 2y^2 + 3xy = 12$ $y^2 - 3y = 4$ I try to turn $x^2 + 2y^2 + 3xy$ into $(x + y)^2 + y^2 + xy$ , but it's a dead end from here. Can anyone please help?
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2answers
222 views

Solving homogeneous quaternary quadratic Diophantine equation

Given the equation $w^2+x^2+y^2+z^2=wx+wy+wz+xy+xz+yz$, how does one systematically enumerate all non-negative integer solutions $\{w,x,y,z\}$?
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2answers
249 views

Prove a quadratic form is positive definite

I want to prove - without using eigenvalues- that the quadratic form $$q(x,y)=Ax^2+2Bxy+Cy^2$$ is positive definite iff $A>0$ and $AC-B^2>0$ This exercise was taken from a practice for a ...
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1answer
55 views

Inequality of expectation of a quadratic form

I was reading a proof in a paper. Let $X$ and $Y$ be two possibly correlated $K$-dimensional random vectors. Suppose $\mathrm{E}(YY^T)=I$, where $I$ is an $K\times K$ identity matrix. By ...
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1answer
39 views

Terminology: what is the “generic character” of a ternary quadratic form?

The title says it all: What is the "generic character" of a ternary quadratic form? Motivation: I'm reading a really old paper, and the author refers to this terminology without any further ...
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3answers
59 views

A simple system of equations

I'm trying to refresh my school math knowlegde and have trouble solving a simple system of equations: $\begin{cases} x + xy + y = -3,\\ x - xy + y = 1. \end{cases}$ I derive $y$ from the second: $y ...
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1answer
32 views

Implications of a prime square dividing a binary quadratic form

Let $u,v$ be positive integers with $\gcd(u,v)=1$, let $k\ge 3$ be an odd integer, and fix a prime $p$. Now what are the implications of $p^2 \mid (u^2+kv^2)$? I know implications in certain cases, ...
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2answers
104 views

how many solutions to non-linear simultaneous equations

I'm doing a Lagrange multiplier optimization problem, and I wound up with the following simultaneous equations: $2x + 1 -2\lambda x = 0$ $4y-2 \lambda y = 0$ $6z-2 \lambda z = 0$ $-x^2 - y^2 - z^2 + ...
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0answers
40 views

can someone break this quad formula down for me?

Can someone explain how this person yield the stuff on the right side using quad formula?
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1answer
51 views

Same quadratic forms on $\mathbb R^n$

Let $q$ be an inner product on $\mathbb R^n$ and $Q$ be its matrix expressed in the canonical basis of $\mathbb R^n$. Assume that the group $$SO(q)=\{A\in M_n(\mathbb R) \ | \ A^TQA=Q\}$$ of ...
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1answer
168 views

Discriminant of a ternary quadratic form

What is the discriminant of a ternary quadratic form $x^2-y^2+z^2-2xy+4yz-6xz$? The answer says, first make it $a_{11}x^2+a_{22}y^2+a_{33}z^2+2a_{12}xy+2a_{23}yz+2a_{13}xz$, and then the discriminant ...
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1answer
91 views

Quadratic forms — rank of matrix

Assume that $M$ is the matrix of some quadratic form (over any field of characteristic not $2$) and set $$Q(\overline{x})=\overline{x}^tM\overline{x}$$ We can replace $M$ by the symmetric matrix ...
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1answer
49 views

Polynomial and super-symmetric tensor

A quadratic function uniquely determines a symmetric matrix. Ok that’s easy. Now a homogeneous polynomial function $f(x)$ also uniquely determines a super-symmetric tensor. My question is how do I ...
4
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3answers
201 views

A conjecture on products/composition of Pell forms

Based on a few brute-force calculations, I've formulated the following. Conjecture. Let $x,y,u,v,p,q,a,b,c \ge 2$ be integers such that $$ (x^2+ay^2)(u^2+bv^2) = p^2+cq^2, $$ and write \begin{align} ...
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2answers
67 views

Divisors of numbers of the form $a^2+2b^2$ with $\gcd(a,b)=1$

Let's say I have a number $n$ which can be written as $a^2+2b^2$ for integers $a,b$. By Fermat/Euler/etc., I know that the primes dividing the squarefree kernel of $n$ cannot be congruent to $5$ or ...
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1answer
39 views

One Sheeted Hyperboloid

Putting this into Wolfram Alpha, I saw that it is a one-sheeted hyperboloid: $$2x^2 - 4xz + z^2 - 4yz = 4$$ Would someone be able to explain how to prove this mathematically? I thought this surface ...
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1answer
19 views

Find $\alpha , \beta$ s.t. $\forall s_i\in\mathbb{Z} ,\frac{\alpha^2}{\beta}\neq\frac{(s_1-s_2)^2+(s_3-s_4)^2+…}{(s_1+s_2)^2+(s_3+s_4)^2+…}$

Let us assume that $\alpha,\beta , s_i\in\mathbb{Z}$ , for $i=1,...,8$. is it possible to choose $\alpha,\beta$ such that for all $s_i\in\mathbb{Z}$ the following equation is $never$ ...
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3answers
151 views

Find $m_1 , m_2,m_3,m_4\in\mathbb{Q}$ s.t. $\forall a_k,b_k\in\mathbb Z,\,m_1(a_1^2+a_2^2)+m_2(a_3^2+a_4^2)\neq m_3(b_1^2+b_2^2)+m_4(b_3^2+b_4^2)$

Let us assume that $a_1 , a_2 , a_3 ,a_4,b_1,b_2,b_3,b_4\in\mathbb{Z}$. If $m_1 , m_2,m_3,m_4\in\mathbb{Q}$, then how can I choose $m_1,m_2,m_3,m_4$, such that the following equation is $never$ ...
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6answers
1k views

what would be the way to solve a system of equations like this one?

Solve: $xy=-30$ $x+y=13$ {15, -2} is a particular solution, but, how would I know if is the only solution, or what would be the way to solve this without "guessing" ?
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1answer
35 views

Approximate root of $\alpha x - \beta y$ over $\mathbb Z$ except origin

Consider the polynomial $ f(x,y) = \alpha x^2 - \beta y^2 $ Prove or disprove: For any choice of $\alpha, \beta \in \mathbb R_0^+$, the polynomial $f$ gets arbitrarily close to $0$ over $\mathbb Z ...
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0answers
159 views

Reducing a linear algebra expression to quadratic form

I am trying to solve the following exercise for my Machine Learning course. Expand this expression so that there are only quadratic terms: $(\mathbf{x} - \mathbf{\mu})^T \mathbf{\Sigma}^{-1} ...
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1answer
24 views

Determine an orthonormal basis so that $s(v_i, v_j) = 0, 1 \leq i, j \leq 3, i \not= j$

Determine an orthonormal basis $ (v_1, v_2, v_3) $ so that $ s(v_i, v_j) = 0, 1 \leq i, j \leq 3, i \not= j $ $s$ is a symmetrical bilinear form given by the matrix A: $$ A = M_\beta(s) = ...
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1answer
61 views

Expectation of an exponentiated quadratic form

Given a multivariate normal random $n\times 1$ vector $X \sim N(\mu,\Sigma)$, what is the expectation $$\mathbb{E}[exp(X^TAX+b^TX)]$$ where $A$ is a $n\times n$ matrix and $b$ is a n-dimensional ...
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3answers
156 views

“Descent” on binary quadratic forms?

Let's say I have the Diophantine equation $$ x^2+3n^2 = y^2+3z^2. \tag{$\star$} $$ where $n$ is a known integer, and we're trying to determine solutions in integers $x,y,z \ge 1$. Rewrite ($\star$) ...
3
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2answers
79 views

Which integers are representable by $x^2+2y^2+7z^2$?

I have been trying to characterize integers representable by several ternary forms and reached a roadblock with this particular form: $$x^2+2y^2+7z^2$$ Ideally, I am looking for a characterization ...
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0answers
19 views

Distribution of a quadratic form

Let $A$ be a symmetric positive definite matrix, and $x$ a random vector. Suppose we know the distribution of $x^\top A x$. What can we say about the distribution of $x^\top x$?
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1answer
160 views

Zeros of quadratic form of vectors

I have a set of vectors defined as $[\mathbf{v}(x)]_n = e^{jn\pi x}; \quad n = 0 ~\text{to}~ (N-1)$ where $\mathbf{v}$ is an $N \times 1$ vector, $j$ is $\sqrt{-1}$, and $-1 \leq x < 1$. For a ...
2
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1answer
50 views

Using continued fractions to well-approximate a quadratic form?

Continued fractions are the "best rational approximation" of other numbers. For a real number $\alpha$ the continued fraction algorithm produces a sequence of integers $\alpha = [a_1, a_2, \dots, ...
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1answer
51 views

Representation of integers by ternary quadratic form $x^2+y^2-z^2$

Let $Q$ be the ternary quadratic form $Q(x,y,z)=x^2+y^2-z^2$. Since $Q(0,p+1,p)=2p+1$ and $Q(1,p+1,p)=2p+3$, we see that for every integer $k$, the equation $E_k:Q(x,y,z)=k$ always has a solution. Is ...
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2answers
165 views

How can I solve this system of equations? [closed]

Here is a system of equations: $$\begin{cases} x^2 + 10y = 41\\ y^2-2z = 23\\ z^2-6x = 17 \end{cases} $$ What's the value of $x$ and $y$ and $z$?
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3answers
267 views

Solving a system of nonlinear (quadratic) equations

Consider the following system of equations: $$\begin{align} (x + 1)^2 [(p - l)^2 + (q - m)^2] &= (a - l)^2 + (b - m)^2 \\ (x + 1)^2 [(p - a)^2 + (q - b)^2] &= x^2[(a - l)^2 + (b - ...
1
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0answers
184 views

Solving a system of equation and finding the largest possible value of one of the variables

This problem comes from question 5 in the PUMAC Algebra A competition (link here): Suppose $w, x, y, z$ satisfy $$w+x+y+z=25$$ $$wx+wy+wz+xy+xz+yz=2y+2x+193$$ The largest possible value of $w$ can ...
0
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1answer
41 views

Will a 2 by 2 quadratic form be negative definitive if it has repeated eigenvalues which are negative?

Say we have the quadratic form $$ f = x^T Q x \\ Q = \begin{bmatrix}-1 & 0 \\ 0 & -1\end{bmatrix}$$ which has repeated eigenvalues $\lambda = -1$. Will the quadratic form be negative ...
1
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2answers
52 views

A curious question about optimizing a function of 2 variables.

Let $f(x,y)$ be defined and has continuous first and second partials on a domain $D$. Also, let $$A = \frac{\partial^2 f}{\partial x^2} \\ B = \frac{\partial^2{f}}{\partial x \partial y} \\ C = ...
3
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1answer
126 views

Does an isotropic vector always exist for an indefinite quadratic forms?

I have some problem reading some paper. In that paper, the author proved that a quadratic form $Q$ has an isotropic vector(of course, nonzero) by showing $Q$ has both nonnegative and nonpositive ...
3
votes
1answer
143 views

Solution to a System of Quadratic Equations

Problem: Solve for the values of a, b Equation 1: $$(x_1-a)^2+(y_1-b)=r^2$$ Equation 2: $$(x_2-a)^2+(y_2-b)^2=r^2$$ Where, $x_1, x_2, y_1, y_2$ and $r$ are all constant values For the ...
6
votes
1answer
199 views

Expressing a quadratic form, $\mathbf{x}^TA\mathbf{x}$ in terms of $\lVert\mathbf{x}\rVert^2$, $A$

EDIT: This question is actually an attempt to solve this. Please take a look. Let $A$ be a symmetric postive-definite $n\times n$ matrix, i.e. $A\in\mathbb{S}_{++}^{n}$ Also, let ...
1
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1answer
232 views

Find max and min subject to constraint ||x|| = 4

$Q(x,y)=7x^{2}+12xy+12y^{2}$ I only know how to do this is $\|(x,y)\|=1$ If $\|(x,y)\|=1$, the eigenvalues are $16$ and $3$. So obviously $\min=3,\max=16$. I don't know what to do if ...