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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0
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1answer
18 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
139 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
votes
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
117 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
118 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
23 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
40 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
134 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
votes
2answers
69 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 ...
1
vote
0answers
17 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
69 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
42 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
votes
1answer
39 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
votes
2answers
159 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
votes
0answers
45 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
172 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
18 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 ...
1
vote
0answers
75 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
31 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
48 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
55 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 ...
0
votes
0answers
34 views

Rank four quadratic form with trivial discriminant

Is there an example of a field $k$, quadratic form $\varphi$ of rank four, which is anisotropic over $k$, has trivial discriminant and is not a Pfister form? In case of rank six one can use Albert ...
3
votes
1answer
107 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
141 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 ...
0
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0answers
17 views

example of quadratic form

Kindly give me an example of regular quadratic forms $q_1, q_2$ over a field $\mathbb F$ such that $D(q_1)=D(q_2)$ and $d(q_1)=d(q_2)$ but $q_1$ is not isometric to $q_2$, where $D(f)=\{d \in \mathbb ...
1
vote
1answer
110 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 ...
0
votes
2answers
45 views

Symmetric Matrix Quadratic Form

Let $A,B\in\mathbb{M}_{n\times n}(\mathbb{R})$ and $A,B$ are symmetric matrics. Prove that if $\vec{x}^TA\vec{x} = \vec{x}^TB\vec{x}$ $\forall\vec{x}$, then $A=B$. Since $A,B$ are symmetric, they are ...
3
votes
2answers
36 views

Rational quadratic forms

The quadratic form $$10x^2+20y^2+2z^2+4xy-6xz+8yz$$ can be written as $x^TAx$, where A = [ [10,2,-3] , [2,20,4] , [-3,4,2] ] Using diagonalization, this can be written in the form ...
1
vote
1answer
63 views

Projecting self-adjoint operator onto closed subspace

Let $H$ be a complex Hilbert space and let $(Q, D(Q))$ be a closed, densely defined, positive semidefinite, Hermitian quadratic form on $H$. (That is, $D(Q)$ is a dense subspace of $H$, $Q$ maps ...
2
votes
1answer
56 views

When does $x^TAx + c^Tx$ have a global minimum?

This question is closely related to my last question about extended quadratic forms. I figured out a nice criterion, when $$f : \mathbb R^n \rightarrow \mathbb R$$ $$f(x) = x^TAx + c^Tx$$ has a ...
1
vote
1answer
45 views

Rescaling of Ternary quadratic forms

I was reading about the Hilbert residue symbol, and the discussion of it starts out with the assumption that we can reformat any ternary quadratic form over the integers into the form ...
1
vote
3answers
150 views

Solving $a^2+3b^2=c^2$

I'm looking for how to solve the equation $a^2+3b^2=c^2$ where $a,b,c$ are integers and $b$ is even, I'm looking for the algorithm used to solve this kind of equations, not just the solution. Regards ...
1
vote
1answer
78 views

Signed determinant of quadratic forms over Q_p

Let $W(k)$ be the Witt-Ring of the field $k$. in this script http://math.uga.edu/~pete/quadraticforms2.pdf at the bottom of page 2 the signed determinant is introduced by $d^\pm (q) = ...
3
votes
1answer
114 views

rank of quadrics

Consider the quadric $xw-yz$ in $\mathbf{P}^3$ (all over $\mathbf{C}$), and the Klein quadric $x_0 x_5+x_1 x_4+x_2 x_3$ in $\mathbf{P}^5$. I want to determine the rank of these quadrics. For the first ...
1
vote
2answers
54 views

Quaternary Quadratic Forms

What is a Quaternary Quadratic Form? I've looked for a definition online and cannot find a precise clear definition. I am not taking a course. Just reading about quadratic forms. Thank you.
2
votes
1answer
67 views

numeric solutions on quadric surfaces

Maybe it's a trivial thing, but I can't seem to find solution I'm looking for. I need to find a parametric solution to the following equation ($\mathbf{A}$ is positive definite): $$ ...
1
vote
1answer
50 views

Primes of the form $x^2+ny^2$ where $n\equiv 1\pmod{4}$ is a squarefree number

Let $n\equiv1\pmod{4}$ be a squarefree number and $p\equiv1\pmod{4n}$ be a prime number. Does there exist $x,y\in\mathbb{N}$ such that $p=x^2+ny^2$?
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votes
0answers
49 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 = ...
1
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1answer
79 views

How do i find a signature of a quadratic form? Also how do i represent a quadratic form as a sum/difference of squares?

For example given $(x,y,z,t) = xy+ y^2+ yz+z^2+zt$ How do i represent it as a sum and difference of squares (i.e. in the form $\sum a_iA_i^2$) and how do i find its trace? Or if i have a quadratic ...
0
votes
4answers
41 views

Finding the three unknowns

Can someone show me the steps to finding the three unknowns of these two equations. $$-a-bx+cx^2 = x^2+2x+1$$ The answers are $a=\ ...\ $, $b=\ ...\ $, and $c=\ ...$ , but I can't see how they ...
1
vote
3answers
67 views

How can I show the complete symmetric quadratic form has no zeros?

The quadratic complete symmetric homogeneous polynomial in $n$ variables $t_1,\ldots,t_n$ is defined to be $$h_2(t_1,\ldots,t_n) := \sum_{1 \leq j \leq k \leq n} t_j t_k = \sum_{j=1}^n t_j^2 + ...
1
vote
0answers
37 views

Classification of quadrtic forms over Q_p

I need some one to recap the topic with me and correct me when i am wrong. There are basically just a few questions at the end,but its important to also show what i know and not just what i dont. ...
3
votes
1answer
61 views

Question regarding quadratic form exercise in Hoffman Kunze

In the book the quadratic form associated with a bilinear form f is defined as $q(\alpha)=f(\alpha,\alpha)$. Then, if $U$ is a linear operator on $\mathbb R^2$ an operator $U^\dagger$ on the space of ...
1
vote
1answer
59 views

A particular quadratic minimization problem

Given $n^2$ constants $a_{11},a_{12},\ldots,a_{1n},a_{21},\ldots,a_{nn}$ and $n^2$ non-negative variables $x_{11},x_{12},\ldots,x_{1n},x_{21},\ldots,x_{nn}$. Find the minimum value of $$\sum_{i=1}^n ...
0
votes
0answers
33 views

Show that a quadratic form is bounded

I have a question (I hope I can formulate it adequately) concerning quadratic forms: If I have a quadratic form $$ T(x_1,x_2)=x^TQx, x=(x_1,x_2)^T $$ which is bounded to above, i.e. ...
0
votes
2answers
27 views

Positive definit quadratic form

I have a very elementar question... If I have a quadratic form $$ T(x_1,x_2)=x^TQx $$ and $Q$ is a positive definit symmetric 2x2-matrix, then does this mean that the quadratic form is positive ...
0
votes
1answer
21 views

Transformation of quadratic form

I've got the following quadratic form $T(x_1,x_2)=x^TQx$, with $$ x=\begin{pmatrix}x_1\\x_2\end{pmatrix}, Q=\begin{pmatrix}\frac{1}{2}(m_1+m_2)L_1^2 & \frac{1}{2} ...
2
votes
1answer
131 views

Quadratic Diophantine Equation $x^2 + axy + y^2 = z^2$

I have been reading about this quadratic Diophantine equation of the form $x^2 + axy + y^2 = z^2$ where x, y, z are integers to be solved and a is a given integer. All integral solutions are given ...
0
votes
0answers
41 views

On modular group and quadratic forms

Let $\Gamma$ be the modular group, is the group of linear transformations of the upper half of the complex plane. Let $\mathbb Q_{N^2{d_K}}/\Gamma$ (the group of positive definite primitive quadratic ...
1
vote
1answer
97 views

Condition for the identity $(a-b)(a+b)=a^2-b^2$

The identity $(a-b)(a+b)=a^2-b^2$ holds true for what condition ???? I tried using real numbers and imaginary numbers but seems like it holds everywhere. Some say... only for ($a>b$) but I don't ...