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

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1answer
19 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 ...
0
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1answer
57 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} + ...
3
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0answers
39 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 ...
0
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1answer
29 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 ...
0
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0answers
18 views

Reduce to diagonal form.

Problem is to reduce $5X^2+3Z^2+4XY-4YZ+6ZX$ into diagonal form over $\mathbb{R}$. With my knowledge, We ned to make a non-singular variable transformation so that above form comes into a form like ...
0
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0answers
19 views

Finding discriminant of this quadratic form

My question is what are the discriminants of $X^2+Y^2 $ and $X^2-Y^2$ over $\mathbb{R}$ and $\mathbb{C}$ and why? It should be $1$ and $-1$ respectively over $\mathbb{R}$. But shouldn't they be same ...
0
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0answers
28 views

Proving $\dim(W)+\dim(W^\perp)=\dim(V)$

I have to prove if $V$ is non-degenerate, then for any subspace $W$ of $V$, $1)$ $\dim(W)+\dim(W^\perp)=\dim(V)$, $2)$ $W^{\perp\perp}=W$ $3)$ $\operatorname{rad}(W)=W\cap W^\perp$ I was doing ...
1
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1answer
39 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 ...
2
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1answer
87 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 ...
0
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2answers
30 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 ...
0
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0answers
14 views

Prove that every reproducing kernel is a positive matrix (and vice versa)

Let $\mathcal{H}$ be a functional hilbert space (defined over a set $S$) with a reproducing kernel K. Prove that: a) $K$ is a positive matrix means the queadtric form is positive, i.e ...
1
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3answers
33 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?
0
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2answers
98 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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0answers
51 views

Index of inertia for quadratic forms

How to find the indies of inertia and canonical quadratic form of: $ f= 2x_2^2 -x_3^2 + 2x_1x_2 - 4x_1x_3 $ so I guess our matrix would look like: $\begin{pmatrix} 0 & 1 & -2 \\ 1 & 2 ...
0
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2answers
44 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 ...
1
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1answer
36 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 ...
2
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1answer
30 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 ...
0
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3answers
52 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 ...
0
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1answer
23 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, ...
1
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2answers
34 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
29 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?
4
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1answer
49 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 ...
0
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1answer
20 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 ...
0
votes
1answer
25 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 ...
1
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1answer
38 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
votes
3answers
166 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} ...
1
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2answers
51 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 ...
1
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1answer
16 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 ...
0
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1answer
14 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 $never$ ...
3
votes
3answers
130 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 $never$ ...
10
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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" ?
1
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0answers
102 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} ...
0
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0answers
53 views

Proof of Fisher-Cochran's theorem

$dim(E)=n$ We have $u_1, u_2..., u_p$ self-adjoint operators which belong to $E$ $(i)$ : $rk(u_1)+...+rk(u_p)=n$ $(ii)$ : $q_1(x)+...q_p(x)=x.x$ with $q_i$ the quadratic form $q_i(x)=u_i(x).x$ for ...
0
votes
1answer
21 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) = ...
0
votes
1answer
30 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 ...
2
votes
3answers
116 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
68 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
15 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 $x^\top x$?
0
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1answer
57 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
votes
1answer
35 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, ...
0
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1answer
35 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 ...
4
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2answers
152 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 $z$?
0
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1answer
100 views

How can I solve this equations? [closed]

Here is a system of equations: $$\left\{ \begin{eqnarray} x^2+2y^2-2yz=100 \\ 2xy-z^2=100 \end{eqnarray} \right. $$ What's the value of $x$ and $y$? According to my math teacher it's a very hard ...
0
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0answers
37 views

Show that for any given d<0, the primitive positive definite quadratic forms of discriminant d all have the same number of automorphs.

Show that for any given d<0, the primitive positive definite quadratic forms of discriminant d all have the same number of automorphs. I think we should let f(x,y) = $ax^{^{2}} + bxy + ...
0
votes
3answers
160 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 - ...
0
votes
0answers
15 views

On discrete subgroups of modular group and quadratic forms

I'm trying to work my way through a couple of papers on product formulae associated with certain modular forms. In "Borcherds Products Associated with Certain Thompson Series" by Chang Heon Kim, the ...
2
votes
0answers
64 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
votes
1answer
30 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
43 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
votes
1answer
46 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 ...