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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16 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
19 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 ...
3
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0answers
31 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 ...
5
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2answers
101 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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0answers
42 views

A question on constructing subharmonic rational functions

Suppose three functions $x\leq y\leq z$ satisfy $x+y+z=0$ and \begin{equation*} \begin{split} \nabla x=&2x\vec a-y\vec b-z\vec c,\\ \nabla y=&-x\vec a+2y\vec b-z\vec c,\\ \nabla z=&-x\vec ...
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1answer
29 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
48 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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0answers
22 views

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) = ...
3
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0answers
45 views

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 ...
4
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0answers
26 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
17 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
38 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 ...
2
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1answer
46 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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2answers
56 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, ...
3
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1answer
29 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
56 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 ...
4
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2answers
140 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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0answers
12 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
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2answers
63 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
45 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
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0answers
30 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
32 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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0answers
32 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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0answers
35 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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0answers
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 ...
0
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0answers
29 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
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1answer
33 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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0answers
10 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 ...
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1answer
25 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
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1answer
28 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
57 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
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2answers
30 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
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0answers
68 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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0answers
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\, ...
2
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3answers
97 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
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
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3answers
86 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
19 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
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1answer
97 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
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1answer
20 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 ...
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1answer
23 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 ...
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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 ...
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0answers
11 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
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2answers
63 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 $b^2-4ac<0$.
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1answer
44 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
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1answer
37 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
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0answers
29 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
131 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
46 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
60 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, ...