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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The value at the integral lattice of a positive definite quadratic form is discrete

A (real) quadratic form is a homogeneous plynomial of degree 2 (with real cofficients), in any number of variables. A quadratic form is positive definite if it is takes only nonnegative values. Now ...
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78 views

Cubes of the Form $3x^2\pm xy+5y^2$, with $x,y$ Coprime

Are there any cubes of the form $3x^2\pm xy+5y^2$, with x, y coprime ? Partly inspired by this question. I tried various computer searches of the form $|x|\le10^a$, $|y|\le10^b$ with $a+b=6$, all ...
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2answers
42 views

Definiteness of a Quadratic Form

The problem is as follows: For what values of c is the quadratic form $$Q(x,y) = 3x^2-(5+c)xy+2cy^2$$ positive definite, positive semidefinite, or indefinite? Ok. My approach was to find the ...
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0answers
26 views

A problem on unitary spaces

If $V$ is a unitary space with a hermitian form $\langle,\rangle$ and $v_1,...v_n$ are any $n$ vectors in $V$ then is it true that ${\rm det}(\langle v_i,v_j\rangle)\geq 0$? When does equality hold? ...
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45 views

Finding integers of the form $3x^2 + xy - 5y^2$ where $x$ and $y$ are integers, using diagram via arithmetic progression

So the diagram drawn looks like this: We begin at the edges labeled $3$ and $-5$ because we are using those as the bases for $x$ and $y$, respectively. The way we obtain the values of the 2 ...
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15 views

How to convert the following problem to the standard quadratic programming form

I have the following problem, which I guess it is QP, but I donot know how to convert it to the standard form. ${\rm minimize}\sum_{j=1}^{N}( y_j - (\sum_{i=1}^{N}p_iyi) )^2$ subject to $p_i\geq0$ ...
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13 views

Developping a long quadratic form like $(x-y-z-\mu)^t\Sigma^{-1}(x-y-z-\mu)$

Is there a way to some how develop a long quadratic form ? Maybe something like : $(x-y-z-\mu)^t\Sigma^{-1}(x-y-z-\mu)= (x-\mu)^t\Sigma^{-1}(x-\mu) - (y+z)^t\Sigma^{-1}(y+z)$ or is there another way ...
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2answers
124 views

What's so special about the form $ax^2+2bxy+cy^2$?

Binary quadratic forms are sometimes studied (e.g. by Gauss) in the form $$ax^2+2bx+cy^2$$ In other words, the second coefficient is assumed to be even, and the polynomial is assumed to be ...
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21 views

Reference about quadratic forms with discriminant 1

When I am reading Serre's $A$ $Course$ $In$ $Arithmetic$, Chapter 5, it deals with $quadratic$ $forms$ of some vector space $V$, which can be viewed as an extension of an $abelian$ $group$ $E$ of ...
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1answer
56 views

Equivalent quadratic forms

Two quadratic forms $$Q(x_1, x_2, \dots , x_n) \\ \text{ and } Q'(x_1, x_2, \dots , x_n)$$ are called equivalent $$\Leftrightarrow Q'(x)=Q(Tx), \text{ where } T \in M_n(K), \text{ invertible }$$ ...
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1answer
38 views

A quadratic form over $K-$vector space $V$

Let $K$ a field, $\operatorname{char} K \ne 2$. Definition: A quadratic form over $K$ is a homogeneous polynomial $Q(x_1, x_2, \dots , x_n) \in K[x_1, x_2, \dots , x_n]$ of degree $2$. If ...
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1answer
24 views

Clarification of some doubts: working with the restriction of a quadratic form

Let $q:\mathbb{R^3}\to\mathbb{R}$ such that $$q(x,y,z)=2x^2+3y^2+4xy-2xz.$$ I have to determine rank and signature of $q$, and so far it should be fine: I got $\operatorname{rk}(q)=3$ and ...
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2answers
32 views

If I have a polynomial $x^2(1-m^2) - x2m^2 - (m^2 + 1)$ with a solution at $x = -1$, how do I get the other root

If I have a polynomial $x^2(1-m^2) - x2m^2 - (m^2 + 1)$ with a solution at $x = -1$, then I know I can just take $x^2(1-m^2) - x2m^2 - (m^2 + 1)$ and divide it by $x+1$ to get the other root. In a ...
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39 views

What is a “supplementary subspace”?

Let $Q$ be a quadratic form of vector space $V$ over a field $k$ with characteristic $\neq 2 $, $V^{0}$ be its orthogonal complement. If $U$ is a supplementary subspace of $V^0$ in $V$, then $V = ...
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2answers
28 views

Finding the value of a constant given an equation where the sum of the roots is -3

I am to find the value of h given the equation 3hx^2 - 2x +5xh = 3. The sum of the roots of the polynomial is -3. I am having ...
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36 views

(Fast) eigen decomposition of $DXD$ where $D$ is diagonal, $X$ is symmetric with known eigen decomposition

Assuming that I already know the eigen-decomposition of a real symmetric matrix $X$, is there any way to use it to retrieve efficiently the eigen-decomposition of $DXD$, where $D$ is a diagonal ...
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2answers
130 views

How to reduce a quartic form to a quadratic form with equal roots

Given a polynomial in $n$ variables of the form $$P(x_1,x_2,\dots,x_n)=\left(\sum_{i,j}a_{ij}x_ix_j+\sum_{i}b_{i}x_i+c\right)^2$$ is there a way to find a polynomial also in $n$ variables of degree ...
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1answer
35 views

Quadratic form on Vector Bundle

A quadratic form of a vector space $V$ over a field $\mathbb{F}$ is a bilinear symmetric map $V\times V \rightarrow F$. How does one define a quadratic form over a vector bundle.
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1answer
50 views

Prove these quadractic forms are equivalent over $\mathbb{Z_5}$

Consider the following quadractic forms, defined in the field $\mathbb{Z_5}$, $$q(x, y, z, t) = 2y^2 + z^2 + 2t^2 + 4xy + 2xt + 4yt$$ $$q_0(x, y, z, t) = x^2 + y^2 + z^2 + dt^2$$ Prove they are ...
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Question on positive definiteness of non homogeneous quadratic form

I'm having trouble understanding a proposition from a semidefinite-programming textbook. It goes as follows: Let $Q$ be a quadratic function of $x \in \mathbb{R}^n$ given by $$Q(x) = ...
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quadratic form from nxn matrices to reals ( Tr(A^2) ). I need to find it's signature and rank.

Firstly prove $Tr(A^2)$ defines a quadratic form from the space of $n \times n$ matrices to R. I think you just have to show that $Tr(A B)$ is a bilinear form which seems too easy to be correct or I'm ...
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31 views

Quadratic optimisation with quadratic equality constraints

I would like to solve the following optimisation problem: $\min_{x} (x'Ax)$ subject to $x'Bx = x'Cx = 1$. Where A is symmetric and B and C are diagonal. Does anyone have a suggestion for an ...
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1answer
27 views

The set of positive definite forms in the space of quadratic forms

Let $u_1,...,u_k\in\mathbb{R}^n$ such that there is a non-zero quadratic form $Q$ satisfying $Q(u_i)=1$ for all $i=1,...,k$. Is there a positive definite quadratic form satisfying the same equations? ...
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1answer
40 views

A question about minors of matrices

Let $B_{\bar{i}\bar{i}}$ denote the remnant of a square matrix $B$ after its $i^{th}$ row and $i^{th}$ column have been removed. Now given any vector $v$ is there some natural relation between the ...
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1answer
76 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 matrizes that define $f,g$. It ...
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57 views

Solve the simultaneous equations $(x+y)^2+3y^{2}=7$ and $x+2y(x+1)=5$

Solve this pair of simultaneous equations: $$\begin{cases} (x+y)^2+3y^{2}&\!\!\!\!\!=7, \\[2pt] x+2y\,(x+1)&\!\!\!\!\!=5. \end{cases} $$ I tried expanding the equations and differencing them, ...
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1answer
32 views

equivalence of quadratic forms

Given two Hermitian positive semidefinite matrices $A$ and $B$, under what conditions on these matrices will $x^H A x = x^H B x$ for all vectors $x$? Clearly, we have equivalence when $A=B$, but I ...
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1answer
97 views

Quadratic forms, diagonal form, and whether an orthogonal transformation exists for a matrix,

a) Let $$ \begin{bmatrix} 3 & 2 & -2 \\ 2 & 3 & -2 \\ -2 & -2 & 5 \\ \end{bmatrix} $$ be a quadratic form. Write explicitly an ...
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2answers
142 views

Is every sufficiently large positive integer of the form $ab + ac + bc + 1$?

Is every sufficiently large positive integer $A$ of the form $ab + ac + bc + 1$ where $a,b,c$ are some positive integers larger than some given positive integer $d$ ? How large is sufficiently ...
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15 views

Group actions (congruence subgroups on integral binary quadratic forms)

I would really appreciate some help in computing the representatives for the space $Q_d/\Gamma_0(N)$ where $\Gamma_0(N) < \mathrm{SL}(2,Z)$ is the congruence subgroup at level $N$ and $Q_d$ is the ...
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2answers
81 views

derivative of a symmetric bilinear form (quadratic form version)

Let $A=A^T\in \mathbb R^{k\times k}$ be a nonzero symmetric matrix and define $F:\mathbb R^k\to\mathbb R$ by $$f(x):=x^TAx$$ Then why $df(x)\xi=2x^TA\xi$ for $x,\xi\in\mathbb R^k$?
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1answer
50 views

(Though?)Expression Rearranging

I have the following expression $ 2x+3x^2+e^{5x+x^2}=7 $ which I need rearranged in a form of the type $Ye^Y=Z$ with Y a function of x and Z some constant. I have tried the substitution $y=5x+x^2$, ...
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31 views

Objects corresponding to Higher forms

If $Q$ is a quadratic form, then we know there exists matrix $A$ such that $Q=xAx'$ and $Q$ can be expressed as weighted sum of eigenvalues of $A$. If $H$ is a higher order form, then is there an ...
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1answer
43 views

Quadratic form - vector/matrix

I have two very simple (stupid) questions about quadratic forms. Having any matrices $A,B$ and vectors $x,y$ (real/complex, singular/regular, rectangular, infinite size, etc.) with appropriate size ...
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35 views

Albert- Algebras and Traceforms

Im new to the topic so this could be basic nonsense to you. Any Albert-Algebra $A$ has a trace map $T:A \rightarrow k$ and thus one can assign a quadratic form $q_A$ of rank $27$ by setting $q_A(x) = ...
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47 views

Show Equivalence of Binary Quadratic Forms

I've been stuck on these two problems from my problem set for quite a while. Any help would be appreciated! 2)Suppose that $ax^2 + bxy + xy^2$ is equivalent to $Ax^2 + Bxy + Cy^2$. Show that $gcd ...
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14 views

Set of squares in quadratic forms of a given discriminant.

For quadratic forms of negative discriminant, the set of squares is the same as the principal genus $H$ (forms whose values in $Z/DZ$ is the same as that of $x^2 + ny^2$ or $x^2 + xy + ny^2$ where ...
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30 views

Odd dimensional universal quadratic form is isotropic?

For odd dimensional nondegenerate universal form, is it isotropic? All isotropic form is universal, but I wonder reverse case. I try to break it down into single form and even dimensional form but it ...
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1answer
37 views

Converting from Non-basis coordinates to XYZ. Solving system of equations. Error volume

I have multiple points in 3D space. Each point has the distances to 3 points. Those 3 points are: (50,0,0) (0,50,0) (0,0,50) Lets call those distances $dx,dy,dz$ I want to find $x,y,z$ of those ...
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Can inequality $-1<(x-\tfrac{1}{2})^2 - 3 (y-\tfrac{1}{2})^2 < 1$ be solved with continued fractions?

It's known at Pell's equation $x^2 - 3 y^2 = 1$ can be solved using the periodic continued fraction expansion of $\sqrt{3}= [1;\overline{1,2}]$. Eventually we get convergents $\tfrac{p}{q} \approx ...
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1answer
30 views

Characterization of integral quadratic forms representing the same numbers? [duplicate]

Is there a simple characterization of integral quadratic forms that represent the same numbers? I know that if two quadratic forms are in the same $GL_n(\mathbb{Z})$-orbit then they represent the ...
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1answer
31 views

What is the solution for this quadratic program?

Given scalars $p_1\geq p_2\geq \cdots \geq p_r > 0$, can we find a solution for following problem? \begin{align} \text{minimize} & & & \sum_{j=1}^{r} p_j (1-t_j)^2 \\ \text{s.t.} \\ ...
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1answer
153 views

Solve nolinear system of equaion with c/c++ [closed]

My system of equation is like this: (x-a1)^2 + (y-b1)^2 = c1 (x-a2)^2 + (y-b2)^2 = c2 I know it is simple using matlab: ...
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2answers
35 views

Quadratic Equations GRE Quants

It would be very useful if someone can give me an answer to this question with a proper explanation. One of the factors of the equation $x^2 +9x + c$ is $(x+11)$, where $c$ is a constant. Which of ...
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82 views

A Tale of Two Quadratic Identities (Pell-like)

Question is at the end. Let all variables be integers. For some constants $a,b,c,d$, assume we have initial solution {$m,n$} to, $$a m^2 + b m n + c n^2 = d\tag{1}$$ Identity 1: $$a x^2 + b x y + ...
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42 views

Theorem for Equal Sums of Like Powers $x_1^8+x_2^8+x_3^8+\dots$

Kindly see the question at the end of post. Solutions to the system of three equations, $$\begin{aligned} a^2+b^2+c^2+d^2\, &= e^2+f^2+g^2+h^2\\ a^4+b^4+c^4+d^4\, &= e^4+f^4+g^4+h^4\\ abcd\, ...
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3answers
131 views

Sinha's Theorem for Equal Sums of Like Powers $x_1^7+x_2^7+x_3^7+\dots$

Sinha’s theorem can be stated as, excluding the trivial case $c = 0$, if, $$(a+3c)^k + (b+3c)^k + (a+b-2c)^k = (c+d)^k + (c+e)^k + (-2c+d+e)^k\tag{1} $$ for $\color{blue}{\text{both}}$ $k = 2,4$ ...
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1answer
24 views

Is the sums of sqaures (without zero) a multiplicative group?

I'm taking a course in quadratic forms at the moment. There is no text book or notes so all I have to go from is what the teacher writes on the blackboard. We had the following lemma some weeks ago: $ ...
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3answers
91 views

For what $a,b$ such that $ax^2+by^2 = z^2$?

This post made me think about this question. What is the criterion on positive integer $a,b$ such that, $$ax^2+by^2 = z^2$$ can be solved in positive integers $x,y,z$? (Three broad classes are: 1) ...
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1answer
23 views

Show that $x \mapsto \left( x^{\top} \sigma x , -\mu^{\top}x \right)^{\top}$ transforms a given set into a convex set.

Let's say you have a covariance matrix $\sigma$ and a vector of expected returns $\mu$. Basically $\sigma$ is a symmetric matrix with positive eigenvalues and we can safely assume that $\mu$ is just a ...