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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4
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2answers
133 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 ...
0
votes
0answers
8 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 ...
2
votes
2answers
56 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 $x,\xi\in\mathbb R^k$?
4
votes
1answer
42 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 $y=5x+x^2$, ...
2
votes
0answers
29 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 ...
1
vote
1answer
29 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 ...
1
vote
0answers
26 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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vote
0answers
26 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 ...
0
votes
0answers
13 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 ...
0
votes
0answers
27 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 ...
0
votes
1answer
29 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 ...
0
votes
0answers
8 views

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 ...
0
votes
1answer
23 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 ...
2
votes
1answer
25 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.} \\ ...
0
votes
1answer
38 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: ...
0
votes
2answers
28 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 ...
3
votes
0answers
60 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 + ...
9
votes
0answers
26 views

Intersection of two hyperplanes

$G$ and $H$ are hyperplanes in $\mathbb{P}_n$ with coordinates $g=(g_0, \ldots, g_n)$, $h=(h_0, \ldots, h_n)$. How can I find a symmetric matrix $A_Q$ of a quadric $Q$ with $ Q = G \cap H$, where ...
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vote
0answers
38 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\, ...
2
votes
3answers
90 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 $\color{blue}{\text{both}}$ $k = 2,4$ ...
1
vote
1answer
23 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: $ ...
4
votes
3answers
85 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) ...
0
votes
1answer
18 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 ...
0
votes
1answer
95 views

Classify the surface: $x^2+y^2-z^2+2xy-2xz-2yz-y=0$

I need to determine what shape the surface is and justify it. Now wolfram alpha tells me that this particular surface is a hyperbolic paraboloid which has the general form: $\alpha x^2-\beta ...
0
votes
1answer
12 views

Is there a way to analytically find a stationary point along an arbitrary line in a multivariable quadratic function?

Let's say I'm working with a quadratic function with an equation of $f(\mathbf{x}) = \frac{1}{2}\mathbf{x}^TA\mathbf{x} - b^T\mathbf{x}$. Now, let's take a direction $\mathbf{p}$ and transform the ...
0
votes
1answer
22 views

Composition of binary quadratic forms as matrix operations

It is easy to see that any binary quadratic form $a^2 + 2bxy + cy^2$ is the same as $XAX^T$ where $X = [x, y]$ and $A = \begin{bmatrix}a & b\\b & c\end{bmatrix}.$ The composition of two ...
1
vote
1answer
25 views

Show that quadratic form $z_0^2+z_1^2+z_2^2$ is invariant under $SO_3(\mathbb{R})$

Let $z=(z_0,z_1,z_2)$. We thus have $z_0^2+z_1^2+z_2^2 = z^Tz=||z||$. Showing invariant means is this what I need to show: $$\forall A\in SO_3(\mathbb{R}), \ ||Az||=||z||?$$ But this is clear from ...
0
votes
0answers
10 views

Solution to a particular quadratic vector equation

I have to solve for $y$ from a quadratic vector equation of the form $$y^{T}Py + 2q^{T}y = 0,$$ where $P \in \mathbf{R}^{n \times n}$ is positive semidefinite, and $q \in \mathbf{R}^{n}$. I got some ...
0
votes
2answers
56 views

Minimum value of a positive definite binary quadratic form along integers

Is there a formula for the least non-zero value of $$f(x,y):=ax^2+bxy+cy^2$$ as $x,y$ assume integer values? Here $a,b,c$ are integers with $a,d>0$ and $b^2-4ac<0$.
0
votes
1answer
38 views

Linear Algebra quadratic forms diagonalization

I have a question that reads: Diagonalize the quadratic form $A(x,x) = 2x^2 - 1/2 y^2 -2xy - 4xz$ by completing the squares, and find the change of basis matrix and the new basis in which A will be ...
0
votes
1answer
34 views

Linear Algebra Quadratic Form Diagonalization

I asked this question the other day but I still didn't understand it. Hopefully someone can get through to me this time. I have a question that reads: Diagonalize the quadratic form A(x,y) = 3x^2 - ...
0
votes
0answers
25 views

Addition and Subtraction of Quadratic Forms

Working through matrix algebra, I have to solve the following expression, which is composed of quadratic forms. My question is how do you simplify this expression? Please be as detailed as possible. ...
5
votes
1answer
123 views

An argument from a blog article of Terence Tao

Let $A_1, A_2, A_3, \ldots , A_m$ be positive semi-definite Hermitian matrices and then consider the polynomial $p(z,z_1,z_2,\ldots,z_m) = \det(z+z_1A_1 + z_2A_2 + \cdots+z_mA_m)$ Now Tao argues that ...
0
votes
2answers
44 views

What is the polarization identity?

Hi I am studying stochastic calculus and my professor often mentions "Polarization Identity" but I do not know how it is defined. I tried googling it but could not find the right definition and ...
3
votes
2answers
53 views

Diagonalizing Quadratic Forms. Linear Algebra

I have a question that reads: Diagonalize the quadratic form $A(x,y) = 3x^2 -12xy + 7y^2$ by completing the square. What is diagonalization? Is that when I should find the eigenvector matrix, ...
0
votes
0answers
38 views

Quadratic form and basis

I computed (1). The matrix A can be diagonalized as P^T A P=D where $$D=\left(\begin{array} -6 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & 12 \end{array} \right)$$ ...
3
votes
2answers
89 views

An annoying Pell-like equation related to a binary quadratic form problem

Let $A,B,C,D$ be integers such that $AD-BC= 1 $ and $ A+D = -1 $. Show by elementary means that the Diophantine equation $$\bigl[2Bx + (D-A) y\bigr] ^ 2 + 3y^2 = 4|B|$$ has an integer ...
0
votes
1answer
47 views

Finding a matrix $P$ such that $P^TAP$ is diagonal

I'm kind of confused on some linear algebra. On a previous question I was given a quadratic form and I had to find a matrix $P$ such that $P^TAP$ is diagonal. I did this by using a suitable change of ...
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votes
0answers
22 views

determinant of the covariance matrix of a normal distribution

Suppose a $p \times 1$ vector $x \sim N_p(\boldsymbol 0, \boldsymbol \Sigma_1)$. Now, There is another covariance matrix $\boldsymbol \Sigma_2$. We know that $|\boldsymbol \Sigma_2| < |\boldsymbol ...
1
vote
3answers
46 views

How would one solve this system

$$12x^2=6z\\2y=-z\\6x-y=7$$ It's been many years since I've dealt with system equations, and now find myself in need to solve them. I am not quite sure what to do; I am interested in finding $x$ and ...
0
votes
2answers
48 views

Equivalent quadratic form with 4 varibles

Consider two quadratic forms: $Q(x,y,z,w)=x^{2}+y^{2}+z^{2}+bw^{2}$ and $P(x,y,z,w)=x^{2}+y^{2}+czw$. For what type of values of $b$ & $c$ (real or complex or negative or positive or zero) $P$ ...
0
votes
1answer
29 views

If $x\in \mathbb{R}^n$ and is a unit vector, why is $\sum\limits_{j,k=1}^n |x_j||x_k| < n^2$?

This is an excerpt of a larger proof: Other pertinent information: $A$ is a positive definite $n \times n$ matrix The set $C$ is the unit sphere I don't get the last inequality: $\gamma \sum ...
0
votes
1answer
25 views

Help explain the geometry being described in this paragraph

In A First Course in Optimization Theory by Sundaram I read on page 51: Given a quadratic form $A$ and any $t\in\mathbb R$, we have $(tx)'A(tx)=t^2x'Ax$, so the quadratic form has the same sign ...
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vote
0answers
22 views

Reduce degree of a high degree unconstrained binary term to quadratic unconstrained binary term

I'm working on a optimization project, in this project I have to convert higher order unconstrained binary polynomial to quadratic unconstrained binary polynomial. Can anyone give me a hint of how to ...
0
votes
1answer
26 views

Signature of a quadratic form

Let $P$ be a $n$_degree polynomial over $\mathbb{R}$. Let the quadratic form on the vector space of $n$_degree polynomials: $$H(P)=\int_0^\infty e^{-x^2}P(x)P(-x)dx$$ What is the signature of $H$? ...
1
vote
1answer
35 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 ...
5
votes
2answers
82 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
votes
1answer
32 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? ...
1
vote
1answer
53 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. ...
1
vote
1answer
64 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} ...